Coloring Eulerian triangulations of the Klein bottle

Transcription

1 Coloring Eulerian triangulations of the Klein bottle Daniel Král Bojan Mohar Atsuhiro Nakamoto Ondřej Pangrác Yusuke Suzuki Abstract We sho that an Eulerian triangulation of the Klein bottle has chromatic number equal to six if and only if it contains a complete graph of order six, and it is 5-colorable, otherise. As a consequence of our proof, e derie that eery Eulerian triangulation of the Klein bottle ith faceidth at least four is 5-colorable. 1 Introduction A graph is said to be Eulerian if all of its ertices hae een degree. In this paper e study colorings of Eulerian triangulations. These are triangulations of closed surfaces hose graph is Eulerian. The celebrated Four Color Theorem [2, 25] asserts that eery planar graph, in particular, eery triangulation of the plane, is 4-colorable. Hoeer, only a handful of planar triangulations are 3-colorable, and it is relatiely easy to see that a planar triangulation is 3-colorable if and only if it is Eulerian [26]. Institute for Theoretical Computer Science, Faculty of Mathematics and Physics, Charles Uniersity, Malostranské náměstí 25, Prague, Czech Republic. kral@kam.mff.cuni.cz. The Institute for Theoretical Computer Science (ITI) is supported by the Ministry of Education of the Czech Republic as project 1M0545. Department of Mathematics, Simon Fraser Uniersity, Burnaby, BC, V5A 1S6, and Department of Mathematics, Uniersity of Ljubljana, Jadranska 19, 1000 Ljubljana, Sloenia. mohar@sfu.ca. Department of Mathematics, Faculty of Education and Human Sciences, Yokohama National Uniersity, 79-2 Tokiadai, Hodogaya-ku, Yokohama , Japan. nakamoto@edhs.ynu.ac.jp. Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles Uniersity, Malostranské náměstí 25, Prague, Czech Republic. pangrac@kam.mff.cuni.cz. Tsuruoka National College of Technology, Tsuruoka, Yamagata , Japan. y-suzuki@tsuruoka-nct.ac.jp. 1

2 Eulerian triangulations are interesting because of their tight connection to 3-colorability of planar graphs. From the algorithmic point of ie, it is triial to decide hether a planar graph is 4-colorable, and an algorithm based on the proof of the Four Color Theorem [24] finds a 4-coloring of a gien planar graph in quadratic time. On the other hand, it is NP-complete to decide hether a planar graph is 3-colorable [12]. Knoing this, it is surprising that a planar graph is 3-colorable if and only if it is a subgraph of an Eulerian triangulation. This fact has appeared in [16] and [18]. Other conditions for 3-colorability of plane graphs can be found e.g. in [7]. In this paper e study Eulerian triangulations of more general surfaces, in particular those of the Klein bottle. We refer the reader for a detailed introduction into graph embeddings on surfaces to a recent monograph [21] and e proide here only the basic results that e need in our further exposition. Going off the plane, the situation concerning chromatic properties of Eulerian triangulations changes drastically. For example, the complete graph K 7 of order seen forms an Eulerian triangulation of the torus and yet, it needs seen colors to be colored. Een more is true. It can be shon that eery graph is a subgraph of some Eulerian triangulation, so the study of chromatic properties of Eulerian triangulations of arbitrary surfaces is as hard as the same problem for arbitrary graphs. Hoeer, there are to directions, here one can make some breakthrough. One is to study locally planar graphs, those ith large edge-idth (see [21] for more details), and the other possibility is to restrict to a fixed surface. Eulerian triangulations of the projectie plane are still ell understood. A set C of ertices of a triangulation T is called a color class if eery face of T has precisely one ertex in C. If C is a color class in the Eulerian triangulation T, then T U is an een-faced map, i.e. an embedded graph hose faces all hae een size. Fisk [11] proed that any Eulerian triangulation of the projectie plane contains a color class. Mohar [19] extended this result by proing: Theorem 1 (Mohar [19]). Let T be an Eulerian triangulation of the projectie plane, and let C be a color class in T. Then the chromatic number χ(t) of T is equal to 3 if and only if T C is bipartite. If T C is non-bipartite and contains a subgraph hich is a quadrangulation of the projectie plane, then χ(t) = 5; otherise χ(t) = 4. Moreoer, gien T, one can find a color class in T and determine χ(t) in polynomial time. The proof of Theorem 1 is based on a discoery of Youngs [31] that quadrangulations of the projectie plane are either bipartite or 4-chromatic, but their chromatic number is neer equal to 3. It also uses an extension by Gimbel and Thomassen [13] ho proed that a graph of girth at least 4 embedded in the projectie plane is 3-colorable if and only if it does not contain a non-bipartite quadrangulation of the projectie plane. As a corollary of Theorem 1 e hae: 2

3 Theorem 2. Eery Eulerian triangulation of the projectie plane is 5-colorable. This result can be proed directly by first obsering that eery projectie planar graph is 6-colorable. Next, Dirac s theorem [1, 8] asserts that the only obstacle for 5-colorability is the presence of a complete graph of order six: the chromatic number of a projectie planar graph is six if and only if it contains a complete graph of order six as a subgraph. Finally, K 6 cannot be a subgraph of an Eulerian triangulation of the projectie plane (this follos from Proposition 4). There are some extensions of these results to more general surfaces. The strongest results in this area hae been obtained by DeVos et al. [9]. Other results stem from the corresponding results about een-faced maps. Note that Eulerian triangulations can be obtained from een-faced maps by placing a single ertex into each face of it and joining it to all the other ertices incident ith that face. An important parameter that quantitatiely measures local planarity of an embedded graph G is the length of a shortest non-contractible cycle in G, hich is called the edge-idth of G. As an example, let us mention the folloing result of Hutchinson [14] (cf. also [9] for a strengthening of this result). For eery genus g, there exists an integer r(g) such that eery graph embedded on an orientable surface of genus g ith all faces een and ith edge-idth at least r(g) is 3- colorable. The claim triially holds in the plane, since such a graph must be bipartite. For the torus, the original bound of 25 from [14] on r(1) has been improed to 9 by Archdeacon et al. [3]. A recent improement [17] of r(1) to 6 is best possible since the Cayley graph C(Z 13 ; 1, 5) has chromatic number four and can be embedded on the torus ith edge-idth fie [3]. In fact, this Cayley graph is the only obstacle for a triangle-free een-faced map on the torus to be 3-colorable. The same phenomena appear for other classes of graphs: eery Eulerian triangulation of an orientable surface that has sufficiently large edgeidth is 4-colorable [14] and eery graph embedded on a fixed surface (orientable or non-orientable) ith sufficiently large edge-idth is 5-colorable [29]. Note that the assumption in the results of Hutchinson [14] that a surface is orientable is essential since eery non-orientable surface admits quadrangulations of arbitrarily large edge-idth, hose chromatic number is four [3, 9, 20]. Hoeer, such non-3-colorable quadrangulations can be characterized [20]. Similarly, Eulerian triangulations on orientable surfaces haing sufficiently large edge-idth are 4-colorable as proed by Hutchinson, Richter, and Seymour [15]. On the other hand, eery non-orientable surface has non-4-colorable Eulerian triangulations of arbitrarily large edge-idth [3, 9], and they can also be characterized [22]. As colorings of Eulerian triangulations of the projectie plane are ell-understood (cf. Theorem 1), e focus on Eulerian triangulations of the Klein bottle, the next simplest non-orientable surface. Our goal is to characterize non-5-colorable Eulerian triangulations of the Klein bottle. In particular, e proe that an Eu- 3

4 lerian triangulation of the Klein bottle is 5-colorable if and only if it does not contain the complete graph of order six as a subgraph. Let us mention at this point that graphs considered in this paper can hae parallel edges as long as they do not form bigons. As a corollary e proe that eery Eulerian triangulation of the Klein bottle ith edge-idth at least four is 5-colorable. It seems to be more difficult to establish analogous results for 4-colorable Eulerian triangulations of the Klein bottle since there are 5-chromatic Eulerian triangulations ith arbitrarily large edge-idth; in particular, the list of forbidden subgraphs in this case cannot be finite. Our main result is the folloing: Theorem 3. An Eulerian triangulation G of the Klein bottle is 5-colorable if it does not contain a complete graph of order six. If G contains K 6, then it can be obtained from one of the maps T A, T B, T C, T D and T E, hich are depicted in Figures 1, 11, 14 and 16, by adding ertices and edges and its chromatic number is equal to 6. Theorem 3 is a combination of Theorems 28 and 31, hich are proed in Sections 8 and 10, respectiely. 2 Outline of the proof Before e start proing our main result, e ould like to explain the major steps of the proof. First, e realize that it is enough to proe our results for proper triangulations as defined in Section 3. In Section 3, e also define a special type of contraction that can be applied to ertices of degree four. This operation has the property that if it is applied to an Eulerian triangulation, the resulting triangulation is also Eulerian and if the resulting triangulation is 5-colorable, so is the original one. This leads us to the need to analyze Eulerian triangulations ithout contractible ertices. Before e proceed ith further analysis, e realize that unless the Eulerian triangulation of the Klein bottle contains a ertex of degree to, is 6-regular or of a ery special type (all these cases are easy to handle), it contains a ertex of degree four adjacent to a ertex of degree six (Lemma 16). In Section 5, e then analyze Eulerian triangulation ith a non-contractible ertex of degree four adjacent to a ertex of degree six and identify one particular configuration that requires a finer analysis e call such a configuration the resistant configuration and the ertex of degree four inoled in it a resistant ertex. Eulerian triangulations ith the resistant configuration are then analyzed in Section 6. All these results are combined in Section 7 here e proe that eery minimal (under our operation of the contraction) non-5-colorable Eulerian triangulation of the Klein bottle is isomorphic to one of the fie triangulations that are denoted by T A, T B, T C, T D and T E and are introduced later. 4

5 We obsere that each of the fie minimal triangulations contains the complete graph K 6 of order six as a subgraph. Moreoer, there are only four non-isomorphic embeddings of K 6 in such triangulations. We call these embeddings bad. In Section 8, e sho that they are the only embeddings of K 6 that can be contained in an Eulerian triangulation of the Klein bottle. In Section 9, e inert the operation of contraction and introduce expansions of ertices. We sho that if an expansion of a ertex results in a triangulation ith no complete graph of order six (and the original graph contained the complete graph of order six as a subgraph), then the obtained graph is 5-colorable. Hence, eery Eulerian triangulation of the Klein bottle that is not 5-colorable must contain one of the four bad embeddings of K 6. We formally combine the just explained results in Section 10 in hich e state and proe our main result (Theorem 31). 3 Preliminaries In this section, e introduce notation used throughout this paper. Graphs hich e consider hae no loops but they can contain multiple edges. If G is a graph and W a subset of its ertices, then G[W] is the subgraph of G induced by W. A k-ertex is a ertex of degree k. Similarly, a k-cycle is a cycle of length k. If G is embedded on a surface and 2-connected, then a k-face is a face bounded by a k-cycle. Most of our proofs deal ith graphs on surfaces. We refer to the monograph [21] for definitions related to graphs embedded on surfaces not mentioned here. If G is such a graph and C = 1... l is a non-contractible l-cycle in G, e can cut the surface along C. This decreases the genus of the surface. In the obtained graph G, the cycle C is either replaced by to copies C and C of the original cycle C or by a 2l-cycle C. In the former case, the ertices of C ill alays be denoted by 1... l and the ertices of C by 1... l. In the latter case, the ertices of C are denoted by 1... l 1... l. The reader is referred for an example to Figure 5 here the Klein bottle as cut along a one-sided cycle u 0. Note that the star in the figure denotes a cross-cap. Let us no introduce some additional notation used in our figures. The regions of a graph that are faces are alays filled ith the hite color and those that can contain additional ertices and edges of G ith the gray color. A triangulation of a surface is an embedding of a loopless graph such that each face is a 3-face and an Eulerian triangulation of the surface is a triangulation in hich all ertices hae een degrees. A triangulation is proper if it has no contractible cycles of length to, eery contractible 3-cycle is facial, and eery 4-cycle bounds a region that contains at most one ertex of G. Besides Eulerian triangulations, e are also interested in near-triangulations and Eulerian near-triangulations. An embedding of a graph G in a surface is a near-triangulation if all its faces are 3-faces ith a possible exception of a single 5

6 distinguished face. An Eulerian near-triangulation is a near-triangulation such that all the ertices not incident ith the distinguished face hae een degrees. An embedding of a graph G is a defectie Eulerian triangulation if it is a triangulation and all the ertices of G hae een degrees except for a single pair of adjacent ertices. Similarly, a 2-defectie Eulerian triangulation is a triangulation ith all the ertices of een degrees except for either to pairs of adjacent ertices or to non-adjacent ertices at distance to. Proposition 4. There is no defectie Eulerian triangulation of the plane. Proof. Suppose that G is a defectie Eulerian triangulation of the plane and let and be to adjacent ertices of odd degree. Remoe the edge from G and consider the dual graph G of G. Since the degrees of all the ertices of G\ are een, G is bipartite. On the other hand, all its ertices hae degree three except for a single ertex of degree four (hich corresponds to the face from hich the edge as remoed). Clearly, such a graph cannot exist. On the other hand, defectie Eulerian triangulations of the projectie plane exist. In the next lemma, e sho that defectie and 2-defectie Eulerian triangulations of the projectie plane are 5-colorable. Lemma 5. Eery defectie or 2-defectie Eulerian triangulation of the projectie plane is 5-colorable. Proof. Consider a defectie or a 2-defectie triangulation G of the projectie plane that is not 5-colorable. By Dirac s Theorem [1, 8], G contains a complete graph of order six as a subgraph. Let H be such a subgraph of G. Note that the complete graph of order six has a unique embedding in the projectie plane and this embedding is a triangulation. If G is a defectie triangulation, color the edge joining the ertices of odd degree red. If G is 2-defectie, color the to edges joining ertices of odd degree red or color a to-edge path beteen to such ertices red. Note that G contains at most to red edges and a ertex of G has odd degree if and only if it is incident ith exactly one red edge. We no obsere that it can be assumed ithout loss of generality that G has no separating contractible 3-cycle ith all inner ertices of een degree. Assume that T is a separating 3-cycle of G and let G T be the subgraph of G formed by the ertices and edges lying inside the closed disc bounded by T. By Proposition 4, the degrees of all the ertices of G T are een. Hence, remoing the interior of T yields a defectie or a 2-defectie triangulation G of the projectie plane. Clearly, G is 5-colorable if and only if G is. Since each facial cycle of H bounds a 2-cell, each face of H either bounds a face in G or contains a ertex of odd degree (and thus a red edge) in its interior. On the other hand, eery ertex of H is incident either ith a red edge (that can be an edge of G contained in H) or ith a face containing a red edge in 6

7 its interior. Otherise, ould hae odd degree and ould be incident ith no red edges. Since H contains six ertices and G contains at most to red edges, there are exactly to non-empty faces of H each containing a single red edge. Moreoer, these faces of H are ertex-disjoint. Hoeer, there are no to disjoint facial triangles in the unique embedding of K 6 in the projectie plane. This contradiction completes the proof. Next, e establish a lemma on the existence of special 5-colorings in plane graphs. This lemma straightforardly follos from the Four Color Theorem, but e decided to proide a proof independent of it. Lemma 6. Let G be a plane graph. For eery to non-adjacent ertices 1 and 2 of G, G has a 5-coloring that assigns the ertices 1 and 2 the same color. Proof. Consider a counterexample G of the smallest size. Clearly, such a graph G is connected and simple. By Euler s formula, G has at least three ertices of degree at most fie. In particular, G has a ertex of degree at most fie that is neither 1 and 2. By the choice of G, G has a 5-coloring that assigns the ertices 1 and 2 the same color. If the neighbors of in G are colored ith at most four distinct colors, the coloring can be extended to G hich is impossible. Hence, the degree of is fie and its fie neighbors 1,..., 5 are colored ith mutually distinct colors. By symmetry, e can assume that 5 is colored ith the color of 1 and 2. Let G ij be the subgraph of G induced by the ertices colored ith the color of i or j, i, j {1,...,4}. Since G is plane, the ertices 1 and 3 are not contained in the same component of G 13 or the ertices 2 and 4 are not contained in the same component of G 24. By symmetry, e can assume that the former is the case, and sitch the to colors used in the component of G 13 that contains 3. In this ay, e obtain a 5-coloring of G that assigns the ertices 1 and 2 the same color and such that the neighbors of are colored ith at most four distinct colors. Such a coloring can be extended to. We conclude that G is not a counterexample to the statement of the lemma and thus there is no counterexample at all. We next focus our attention to Eulerian near-triangulations and sho that if the size of the distinguished face is small and G is embedded in the plane, then the parities of degrees of the ertices incident ith the distinguished face are ery restricted. Proposition 7. If G is an Eulerian near-triangulation of the plane ith the distinguished face bounded by a to-cycle 1 2, then the degrees of both 1 and 2 are odd. Proof. By the hand-shaking lemma, the parities of the degrees of 1 and 2 are the same. If they are both een, e obtain by remoing one of the to parallel edges 1 2 a defectie plane Eulerian triangulation hich is impossible by Proposition 4. 7

8 Proposition 8. If G is an Eulerian near-triangulation of the plane ith the distinguished face bounded by a three-cycle 1 2 3, then the degrees of 1, 2 and 3 are een. Proof. By the hand-shaking lemma, either the degrees of all the ertices 1, 2 and 3 are all een or exactly to of them hae odd degree. In the latter case, G is a defectie Eulerian triangulation of the plane hich is impossible by Proposition 4. Arguments similar to those used in the proofs of Propositions 7 and 8 yield the proofs of the next to propositions. We hae decided not to include their (short) proofs. Proposition 9. If G is an Eulerian near-triangulation of the plane ith the distinguished face bounded by a four-cycle , then one of the folloing holds: the degrees of all the ertices 1, 2, 3 and 4 are odd, the degrees of 1 and 3 are een, and those of 2 and 4 are odd, or the degrees of 1 and 3 are odd, and those of 2 and 4 are een. Proposition 10. If G is an Eulerian near-triangulation of the plane ith the distinguished face bounded by a 5-cycle , then there exists an index i such that the degrees of the ertices i and i+1 are odd (the indices are modulo fie) and the degrees of the ertices j, j i, i + 1, are een. Since e are interested in colorings of Eulerian triangulations, e ill also need some tools that allo us to extend precolorings of some of the ertices of a graph to the entire graph. Proposition 11. Let G be a plane Eulerian near-triangulation ith the distinguished face bounded by a k-cycle 1... k, k 5, and let W = { 1,..., k }. Eery precoloring c of the ertices of W that is a proper coloring of G[W] can be extended to a proper 5-coloring of G. Proof. If k 3, then all the ertices 1... k hae mutually distinct colors. Clearly, G is 5-colorable and the colors of such a 5-coloring of G can be permuted to match the colors of 1... k. Suppose no that k = 4. By Proposition 9, G can be completed to an Eulerian triangulation either by adding an edge or a ertex of degree four. Hence, G is 3-colorable. Fix a 3-coloring of G. If the ertices 1,..., 4 should be colored ith four mutually distinct colors, recolor 3 and 4 ith the fourth and the fifth color. If to of the ertices, say 1 and 3 should hae the same color, recolor 1 and 3 ith the fourth color and 4 ith the fifth color. If, in addition, the colors of 2 8

9 and 4 should be the same, also recolor 2 ith the fifth color. In this ay e obtain a coloring that matches the precoloring of 1,..., 4 (up to a permutation of the colors). It remains to consider the case k = 5. By Proposition 10, e can assume that the degrees of 3 and 4 are odd and the degrees of the remaining ertices are een. Consider the plane graph G obtained by adding edges 1 3 and 1 4. G is an Eulerian triangulation of the plane and it is thus 3-colorable. Since all the ertices 2,..., 5 must receie colors different from the color of 1, the colors of 2 and 4 and the colors of 3 and 5 are the same. We no recolor some of the ertices 1,..., 5 to match the colors in the precoloring. Let A i, i = 1,..., 5, be the set of the ertices 1,..., 5 precolored ith the i-th color. Suppose first that the ertices 1,..., 5 are precolored ith three distinct colors. By symmetry, e can assume that A 1 = 1 and A 2 = A 3 = 2. We proceed as follos: the ertex of A 1 keeps its color and the ertices of A 2 and A 3 are recolored ith the fourth and the fifth color, respectiely. If the ertices 1,..., 5 are precolored ith four distinct colors, e may assume that A 1 = A 2 = A 3 = 1, A 4 = 2 and the ertices contained in A 1 and A 2 are adjacent. In this case, the ertices of A 1 A 2 keep their colors, the ertex of A 3 is recolored ith the fourth color and the ertices of A 4 ith the fifth color. Finally, if the ertices 1,..., 5 are precolored ith fie distinct colors, the ertices 1,..., 3 keep their colors and the ertices 4 and 5 are recolored ith the fourth and the fifth colors. Similarly, some special precolorings of plane Eulerian near-triangulations ith the distinguished face of length six or seen can also be extended as e sho in the next three propositions. Note that the ertex 6 is not assumed to be precolored in the next proposition and its color is determined by the extended coloring. Proposition 12. Let G be a plane Eulerian near-triangulation ith the distinguished face bounded by a 6-cycle and let W = { 1,..., 5 }. Eery precoloring c of the ertices of W ith fie colors that is a proper coloring of G[W] such that c( 1 ) = c( 4 ) can be extended to a 5-coloring of the entire graph G. Proof. Let G be an isomorphic copy of G, the ertices bounding the distinguished face of G, and H a plane graph obtained from G and G by identifying the ertices i and i for i = 1,..., 6. Clearly, H is an Eulerian triangulation of the plane and thus it is 3-colorable. Consider the 3-coloring of G that is the restriction of a 3-coloring of H. Recolor the ertices 1 and 4 ith an unused (fourth) color and the ertex 5 ith another unused (fifth) color. If c( i ) = c( 5 ) for i {2, 3}, also recolor such a ertex i ith the fifth color. Note that the colors c( 2 ) and c( 3 ) are distinct from c( 1 ) = c( 4 ). The coloring of 1,..., 5 no matches the precoloring of G[W] (up to a permutation of the colors). 9

10 Proposition 13. Let G be a plane Eulerian near-triangulation ith the distinguished face bounded by a 6-cycle and let W = { 1,..., 5 }. Eery precoloring c of the ertices of W ith fie colors that is a proper coloring of G[W] such that c( 1 ) = c( 3 ) can be extended to a 5-coloring of the entire graph G. Proof. Let G and H be the graphs as in the proof of Proposition 12 and consider the 3-coloring of G obtained by restricting a 3-coloring of H. Recolor the ertices 1 and 3 ith an unused (fourth) color and the ertex 2 ith another unused (fifth) color. If c( i ) = c( 1 ) for i {4, 5}, also recolor such a ertex i ith the fourth color, or if c( i ) = c( 2 ), recolor i ith the fifth color. The coloring of 1,..., 5 no matches the precoloring. Proposition 14. Let G be a plane Eulerian near-triangulation ith the distinguished face bounded by a 7-cycle and let W = { 1,..., 6 }. Eery precoloring c of the ertices of W ith fie colors that is a proper coloring of G[W] such that c( 1 ) = c( 5 ) and c( 2 ) = c( 6 ) can be extended to a 5-coloring of the entire graph G. Proof. As in the preious to proofs, e consider the 3-coloring of G obtained by restricting a 3-coloring of to copies of G pasted along the boundary of the distinguished face. Recolor no the ertices 1 and 5 ith an unused (fourth) color and the ertices 2 and 6 ith another unused (fifth) color. If c( 3 ) = c( 1 ), also recolor 3 ith the fourth color, and if c( 4 ) = c( 2 ), recolor 4 ith the fifth color. The coloring of the ertices 1,..., 6 no matches the precoloring of G[W]. We no introduce a special type of contractions hich ill be used later in the paper. If G is a triangulation ith a ertex of degree four that is adjacent to ertices 1, 2, 3 and 4 (in this cyclic order around ), a triangulation obtained by a contraction of 1 3 is the triangulation G. 1 3 constructed from G in the folloing ay: the edges 1 and 3 are contracted to a ne ertex, the edges 2 1, 2 and 2 3 are identified to a single edge 2 and the edges 4 1, 4 and 4 3 to a single edge 4. If 1 3 and the ertices 1 and 3 are not adjacent, G. 1 3 is again a triangulation. Moreoer, if the original triangulation is Eulerian, then the obtained triangulation is also Eulerian: the degrees of the ertices 2 and 4 decrease by to and the degree of the ertex is equal to the sum of the degrees of the ertices 1 and 3 decreased by four. Similarly, if G is a defectie Eulerian triangulation, then G. 1 3 is also a defectie Eulerian triangulation, and if G is 2-defectie, then G. 1 3 is either Eulerian, defectie or 2-defectie Eulerian triangulation. A ertex of degree four such that G. 1 3 or G. 2 4 is a triangulation is said to be contractible; otherise, is called non-contractible. Note that if is a non-contractible ertex, then 1 = 3 or the ertices 1 and 3 are adjacent, and similarly, 2 = 4 or the ertices 2 and 4 are adjacent. In the case of the Klein 10

11 bottle, e call a non-contractible ertex resistant if the degrees of 1, 2, 3 and 4 are at least six, the cycle 1 3 is a to-sided non-separating 3-cycle and 2 4 is a one-sided non-separating 3-cycle. At the end of this section, let us recall Theorem 3.1 from [6]. The theorem is stated in [6] for simple plane graphs G but the proof readily translates to graphs ith multiple edges as long as there are no 2-faces. Theorem 15 ([6]). Let G be a plane near-triangulation ith the distinguished face bounded by a k-cycle. If all the ertices that are not incident ith the distinguished face hae degree at least six, then G contains at most k 2 /12 + k/2 + 1 ertices. In particular, if k = 6, then G contains at most one ertex that is not incident ith the distinguished face and if k = 8, then G contains at most to such ertices. 4 Degree structure of Eulerian triangulations of the Klein bottle In this section, e first describe Eulerian triangulations of the Klein bottle ith minimum degree at least four that do not contain a 4-ertex adjacent to a 6- ertex. Such triangulations are either 6-regular or of a ery special type: Lemma 16. Let G be an Eulerian triangulation of the Klein bottle ith minimum degree at least four. Then, either G is 6-regular, or G contains a ertex of degree four adjacent to a ertex of degree at most six, or G contains only ertices of degree four and eight, no to ertices of degree four are adjacent and each ertex of degree eight is adjacent to four ertices of degree four. Proof. Assume that neither of the first to cases apply. Let Q be the set of ertices of degree four of G and O the set of ertices of degree at least eight and set q = Q and o = O. Note that all neighbors of each ertex of Q are in the set O by our assumption. Color no the edges incident ith the ertices of degree four by blue. Next, the edges that are contained in a 3-face incident ith a ertex of degree four and that are not blue are colored red. Let b = 4q be the number of blue edges and r the number of red edges. Each 3-face incident ith a ertex of degree contains one red and to blue edges. Since each ertex of degree four is contained in four faces, there are b 3-faces incident ith ertices of degree four. On the other hand, the number of such 3-faces is at most 2r (each red edge is contained in at most such faces). We conclude b 2r. By Euler s formula, the aerage degree of G is six. Therefore, o q and the equality holds if only if all the ertices in O hae degree exactly eight. 11

12 Figure 1: The Eulerian triangulation T A of the Klein bottle. The number of incidences of the ertices in O ith red edges and blue edges is b + 2r 2b = 8q. Hence, the aerage degree of G is at least 4q + 8q + 6s o + q + s 12q + 6s 2q + s = 6 (1) here s is the number of ertices of degree six of G. Since the aerage degree of G is six, all the inequalities b + 2r 2b and (1) are equalities. It follos that o = q and b + 2r = 2b = 8q. Therefore, all the ertices of O hae degree exactly eight, all the edges incident ith ertices of O are red or blue and b = 2r. Since to consecutie edges incident ith a ertex of O cannot be blue, each ertex of O is incident ith four blue and four red edges. In particular, it is not adjacent to a ertex of degree six. Since G is connected, e conclude that G contains no ertices of degree six. The statement of the lemma follos. At the end of this section, e briefly deal ith one of the cases described in Lemma 16. The structure of 6-regular triangulations of the Klein bottle is ellunderstood [23, 28] and in particular, the chromatic number of eery 6-regular triangulation of the Klein bottle has been determined by Sasanuma [27]. The only 6-chromatic 6-regular triangulation of the Klein bottle is the one obtained from the complete graph of order six by adding a perfect matching (it is depicted in Figure 1). We henceforth refer to this triangulation as the triangulation T A. Lemma 17. A 6-regular triangulation G of the Klein bottle is 5-colorable unless G is the triangulation T A depicted in Figure 1. 5 Eulerian triangulations ith non-contractible 4-ertices In this section, e analyze Eulerian triangulations of the Klein bottle that contain a 4-ertex that is neither contractible nor resistant. Those ith resistant ertices are analyzed in the next section. We start ith the simplest case that a 4-ertex is non-contractible since it is contained in a short separating cycle. 12

13 G u 2 G u 2 u 1 u 1 Figure 2: The projectie plane graph obtained by cutting along a one-sided 2- cycle in the proof of Lemma 19. Lemma 18. If G is an Eulerian triangulation of the Klein bottle that contains a surface-separating 2-cycle or 3-cycle, then G is 5-colorable. Proof. Cut the surface along the non-contractible 2- or 3-cycle. In case that the length of the cycle is to, remoe one of the to parallel edges bounding a ne 2-face. In this ay, e obtain to Eulerian triangulations of the projectie plane or to defectie Eulerian triangulations of the projectie plane hich are 5-colorable by Theorem 2 or by Lemma 5. Their colorings can be easily combined to obtain a coloring of G ith fie colors. We no deal ith Eulerian triangulations ith a 4-ertex contained in a cycle of length to. Lemma 19. If G is a proper Eulerian triangulation of the Klein bottle that contains a surface-non-separating 2-cycle such that the degree of is four, then G is 5-colorable. Proof. Let u 1 and u 2 be the to neighbors of different from. The 2-cycle forms either a one-sided or a double-sided non-separating cure in the Klein bottle. We cut the surface along it. If the 2-cycle is one-sided, e obtain the projectie plane graph G depicted in Figure 2. Remoe the ertices and and the edges u 1 and u 2 and identify the ertices and. In this ay, e obtain an Eulerian triangulation of the projectie plane hich is 5-colorable by Theorem 2. Assigning the same colors to the ertices of G yields a proper 5-coloring of G that can be extended to the ertex (since its degree is four). If the 2-cycle is double-sided, e obtain the plane graph G as depicted in Figure 3. By Lemma 6, G {, } has a 5-coloring that assigns the ertices and the same color. Assign no the ertices of G the colors of their counterparts in G and extend the coloring to. In this ay, e obtain a 5- coloring of G. 13

14 G G u 2 u 1 u 1 u 2 Figure 3: The plane graph obtained by cutting along a to-sided 2-cycle in the proof of Lemma 19. G u 2 u 0 u 0 u 1 Figure 4: The notation used in the proof of Lemma 20. Next, e analyze the case that one neighbor of a non-contractible 4-ertex is also a 4-ertex. Lemma 20. Let G be a proper Eulerian triangulation of the Klein bottle ith no contractible ertices. If G contains to adjacent ertices of degree four, then G is 5-colorable. Proof. By Lemma 18, e may assume that G does not contain separating 2- or 3- cycles. Let and be to adjacent ertices of degree four. If or is contained in a non-contractible 2-cycle, then G is 5-colorable by Lemma 19. Otherise, each of and is contained in to non-contractible 3-cycles. Let u 0, u 1 and u 2 be their neighbors as depicted in Figure 4. Note that the homotopies of the cycles u 1 u 2 and u 1 u 2 are the same. If both the cycles u 1 u 2 and u 0 are one-sided, then their homotopies are the same. Cutting the surface along the cycle u 0 yields a projectie plane graph G. Since the cycles u 1 u 2 and u 0 are homotopic, one of the areas bounded by u 0 u 1u 2 and u 0 u 1u 2 is a 2-cell hile the other one contains a cross-cap (hence, e hae obtained the left or the middle configuration depicted in Figure 5). Color the ertices u 0, u 1 and u 2 ith three mutually different colors and color the ertex u 0 ith the same color as u 0. The coloring can be extended to both the graphs 14

15 u 0 u 1 u 0 u 1 u 0 u 1 u 2 u 0 u 2 u 0 u 2 u 0 Figure 5: The projectie plane graph G obtained by cutting along the one-sided cycle u 0 in the proof of Lemma 20. The left or the middle graph is obtained if the cycle u 1 u 2 is one-sided, and the right one if the cycle is to-sided. G u 0 u 1 u 2 u 0 Figure 6: The plane graph G obtained by cutting along the to-sided cycle u 0 in the proof of Lemma 20. contained in the areas bounded by u 0u 1 u 2 and u 0u 1 u 2 : these graphs are either Eulerian triangulations or defectie Eulerian triangulations of the plane or the projectie plane and thus they are 5-colorable by Fie Color Theorem, Theorem 2 or Lemma 5. The constructed 5-coloring of G readily yields a 5-coloring of G. If the cycles u 1 u 2 and u 1 u 2 are double-sided and the cycle u 0 is onesided, e also cut the surface along the cycle u 0. This yields the projectie plane graph G depicted in the right part in Figure 5. Remoe the ertices,, and and the edges u 0 u 1 and u 0 u 2 and identify the ertices u 0 and u 0 to a ertex u 0. The obtained projectie plane triangulation G contains to ertices of odd degree: u 1 and u 2. Note that u 1 and u 2 are adjacent and thus G is a defectie Eulerian triangulation hich is 5-colorable by Lemma 5. The 5-coloring of G can be extended to the ertices and to a 5-coloring of G. It remains to consider the case that the cycles u 1 u 2 and u 1 u 2 are one-sided and the cycle u 0 is double-sided. Let G be the plane graph obtained by cutting along the cycle u 0 (see Figure 6). By Lemma 6, G has a 5-coloring that assigns the ertices u 0 and u 0 the same color. This 5-coloring can be easily extended to and yielding a 5-coloring of G. It remains to analyze one more case hen a 4-ertex is neither contractible nor resistant. We do so in the next lemma. 15

16 G u 4 u 3 G u 4 u 3 u 1 u 3 u 1 u 3 u 2 u 2 Figure 7: The to possible configurations contained in the graph G in the proof of Lemma 21. Lemma 21. Let G be a proper Eulerian triangulation of the Klein bottle ith no contractible ertices. If G contains a ertex of degree four that is not resistant and that is adjacent to a ertex of degree six, then G is 5-colorable. Proof. By Lemmas 18 and 19, e can assume that is not contained in a 2-cycle. In particular, the ertices u 1, u 2, u 3 and u 4 are mutually distinct. By symmetry, e assume that u 1 is the neighbor of of degree six. By the assumption of the lemma, both the cycles u 1 u 3 and u 2 u 4 are one-sided. In particular, one of the neighbors of u 1 is the ertex u 3, and G contains one of the to configurations depicted in Figure 7. Hence, cutting along the cycle u 1 u 3 yields one of the four projectie plane graphs depicted in Figure 8. Note that one of the to shaded areas in the figure is a 2-cell and one contains a cross-cap since the homotopies of the cycles u 1 u 3 and u 2 u 4 are the same. We no construct a 5-coloring of G for each of the four cases depicted in Figure 8. In the first case, the projectie plane graph bounded by u 2 u 3 u 4 is either an Eulerian triangulation or a defectie Eulerian triangulation. Hence, it is 5-colorable by Theorem 2 or Lemma 5. Next, color the ertex u 3 by the same color as u 3 and the ertices u 1 and u 1 ith the same color different from the colors of u 2, u 3 and u 4, and extend the coloring to the graph inside the 4-cycle u 1 u 2u 4 u 3 by Proposition 11. In this ay, e eentually obtain a 5-coloring of G by assigning a color to the ertex. In the second case, let G be the projectie plane graph bounded by the cycle u u 3 u 4. If the ertices 1 and u 3 are not adjacent, remoe the edge 2u 3 and identify the ertices 1 and u 3. In this ay, e obtain an Eulerian triangulation or a defectie Eulerian triangulation of the projectie plane hich is 5-colorable by Theorem 2 or Lemma 5. Next, color the ertices u 1 and u 1 ith a color different from the colors of u 2, u 3 = u 3 and u 4 and extend the obtained coloring to. In this ay, e obtain a 5-coloring of G. We proceed analogously if the ertices 2 and u 2 are not adjacent. Assume no that both the ertices 1 and u 3 and the ertices 2 and u 2 are adjacent (see Figure 9). If the ertices u 4 and 1 are not adjacent, precolor them 16

17 u 1 u 4 u 3 u 1 u 4 u 3 u 1 u 4 u 3 u 3 u 2 u 1 u u 2 u 1 u 3 u 2 u 1 u 1 u 4 u 3 u 3 u 2 u 1 Figure 8: The four projectie plane graphs G that can be obtained by cutting along the cycle u 1 u 3 in the proof of Lemma 21. u 4 u 3 u 4 u 3 u u u 1 u 2 1 Figure 9: The configurations that appear in G in the proof of Lemma 21 in the analysis of the second configuration depicted in Figure 9. 17

18 = = 6 4 = 7 3 = 8 Figure 10: The resistant configuration. Note that 1 = 5, 2 = 6, 3 = 8, and 4 = 7. ith the same color and color the ertices u 2, 2 and u 3 ith mutually distinct colors different from the color of u 4 and 1. Extend this coloring to the graphs inside the 5-cycle u 2 u 4 u and the 4-cycle 1 u 3 2u 2 by Proposition 11. Next, color the ertices u 1 and u 1 ith the same color and obtain a 5-coloring of G by assigning a color to the ertex. Again, e proceed similarly if the ertices 2 and u 4 are not adjacent. If G contains both the edges 1 u 4 and 2 u 4, then the degree of u 4 is seen note that all the faces incident ith u 4 in the graph depicted in Figure 9 are 3-faces and G is a proper triangulation. Hoeer, this case is excluded by our assumption that G is an Eulerian triangulation. The third and the fourth configurations depicted in Figure 8 are symmetric. Hence, e just analyze the third one. Let G be the projectie plane graph bounded by the 4-cycle u 2 u 3 u 4. Add an edge beteen the ertices u 3 and u 4. In this ay, e obtain an Eulerian triangulation, a defectie Eulerian triangulation or a 2-defectie Eulerian triangulation of the projectie plane. In all the cases, G ith the added edge is 5-colorable by Theorem 2 or Lemma 5. Next, color the ertices u 1 and u 1 ith a color different from the colors of u 3, u 2 and u 4 and extend the coloring inside the cycle u 2 u 1 u 3 u 4 by Proposition 11. A 5-coloring of G is then obtained by assigning a color to the ertex. 6 Eulerian triangulations ith resistant ertices In this section, e analyze Eulerian triangulations of the Klein bottle that contain a resistant ertex. If G is such a triangulation, then it is of the form depicted in Figure 10 in hich the pairs of ertices 1 and 5, 2 and 6, 3 and 8, and 4 and 7 are the same. The resistant ertex is the ertex. Throughout this section, e refer to the configuration depicted in Figure 10 as to a resistant configuration and all the arithmetic ith indices of the ertices i, i = 1,..., 8, throughout this section, is modulo eight. When referring to adjacencies beteen a ertex i 18

19 = = 6 4 = 7 3 = 8 Figure 11: The Eulerian triangulation T B of the Klein bottle. and a ertex different from all i, the ertices i are treated as eight distinct ertices, i.e., if the configuration contains an edge 4 but not 7, e say that is adjacent to 4 and not adjacent to 7. Lemma 22. Let G be a proper Eulerian triangulation of the Klein bottle ith a resistant configuration. If G does not contain a ertex of degree four different from, then G is either 5-colorable or it is the triangulation T B depicted in Figure 11. Proof. By Theorem 15, G contains at most to ertices different from and different from all i, i = 1,...,8. If G contains at most one such ertex, then a 5-coloring of G can be obtained as follos: first, color the neighbors of by four mutually distinct colors and then extend the coloring to the remaining (non-adjacent) ertices of G. The same argument orks if G contains to such non-adjacent ertices. Hence, e can assume that G contains adjacent ertices 1 and 2 that are different from and i, i = 1,...,8. Since the degree of both 1 and 2 is at least six, each of them must be adjacent to fie consecutie ertices i,..., i+4. There are four such triangulations G (see Figure 12): to of them are isomorphic to triangulation T B and the remaining ones are not Eulerian. Next, e obsere that each ertex of degree four different from is adjacent to at least to of the ertices 1,..., 8. Lemma 23. Let G be a proper Eulerian triangulation of the Klein bottle ith the resistant configuration. If G contains a non-contractible ertex, then is adjacent to at least to of the ertices 1,..., 8 and such to ertices are consecutie in the cyclic order around. Proof. By Lemma 20, and are not adjacent. Let u 1,..., u 4 be the neighbors of in the cyclic order around. Since is resistant, all the four neighbors of are mutually distinct and G contains non-contractible 3-cycles u 1 u 3 and u 2 u 4. 19

20 = = = = 6 4 = 3 = = 7 3 = = = = = 6 4 = 7 3 = 8 4 = 7 3 = 8 Figure 12: The four triangulations of the Klein bottle obtained in the proof of Lemma

21 G 1 1 = 5 6 G 2 1 = 5 5 G 2 2 = 6 5 G 1 2 = 6 4 = 7 3 = = 7 3 = 8 6 G 2 1 = 5 5 G 1 2 = 6 4 = 7 3 = 8 Figure 13: The three configurations analyzed in the proof of Lemma 24. If neither u 1 nor u 3 is one of the ertices 1,..., 8, the 3-cycle u 1 u 3 must be contractible. Hence, u 1 or u 3 (or both) are one of the ertices 1,..., 8. Similarly, u 2 or u 4 is one of the ertices 1,..., 8. The claim no readily follos. Our first step toards the analysis of the triangulation ith the resistant configuration is excluding the case that G has non-contractible ertex adjacent to ertices i and j ith j i large. Lemma 24. Let G be a proper Eulerian triangulation of the Klein bottle ith a resistant configuration. If G contains a non-contractible ertex that is adjacent to i and j such that j i 2, i 1, i + 1, i + 2, then G is 5-colorable. Proof. By symmetry, it is enough to analyze the case hen is adjacent to one of the folloing pairs of ertices: 1 and 4, 1 and 5, 1 and 6, 3 and 7, and 3 and 8. The cases that is adjacent to 1 and 5, or to 3 and 8 are handled by Lemma 19. In the remaining cases, let G 1 and G 2 be the near-triangulations contained in the shaded areas as in Figure 13. We no proceed ith each of the three configurations separately: is adjacent to 1 and 4. Consider the near-triangulation G 1, add an edge beteen 6 and 8, and identify the ertices 1 and 5. By Lemma 6, the resulting plane graph G 1 has a coloring ith fie colors that assigns the 21

22 y 1 = 5 6 y 1 = 5 5 x 2 = 6 5 x 2 = 6 4 = 7 3 = 8 4 = 7 3 = 8 Figure 14: The Eulerian triangulations T C and T D of the Klein bottle. ertices 4 and 7 the same color. Color the ertices of G ith the colors of their counterparts in G 1. The obtained coloring can be extended to G 2 by Proposition 11 and eentually to the ertex as ell. is adjacent to 1 and 6. Color the ertices 1,..., 8 ith four mutually distinct colors in such a ay that the pairs of the corresponding ertices receie the same colors. Extend the precoloring to G 1 by Proposition 14. This determines the color of. The coloring can no be extend to G 2 by Proposition 11 and eentually to, since the degree of is four. is adjacent to 3 and 7. Precolor the ertices 1,..., 8 ith four mutually distinct colors in such a ay that the pairs of the corresponding ertices receie the same colors. The precoloring can be extend both to G 1 and G 2 by Proposition 12. Since the degree of in G is four, it can be recolored so that its color is the same in both the extensions of the precoloring. At the end, e color the ertex. Next case that e consider is hen there is a non-contractible 4-ertex that has exactly to neighbors in common ith. Lemma 25. Let G be a proper Eulerian triangulation of the Klein bottle ith the resistant configuration and ithout contractible ertices. If G contains a noncontractible ertex that is adjacent to exactly to of the ertices 1,..., 8, then G is 5-colorable or it is the Eulerian triangulation T C or T D depicted in Figure 14. Proof. Let i and j be the neighbors of, and let x and y be the remaining to neighbors of. Since is non-contractible, x is adjacent to the counterpart of i and y to the counterpart of j. If j {i ± 1, i ± 2}, then G is 5-colorable 22

23 y 1 = 5 6 y 1 = 5 5 x 2 = 6 5 x 2 = 6 4 = 7 3 = = 7 3 = y 1 = 5 6 y 1 = 5 5 x 2 = 6 5 x 2 = 6 4 = 7 3 = 8 4 = 7 3 = 8 Figure 15: The four configurations considered in the proof of Lemma 25. by Lemma 24. By symmetry, e can assume that i is equal to 1, 2 or 3, and j {i + 1, i + 2}. If i = 3 and j = 4, then the ertex x cannot be adjacent to 8 and y to 7 (because of simple topological reasons). We conclude that G contains one of the four configurations depicted in Figure 15 (the dashed edges are dran for future reference and are not contained in the configurations). Let us analyze the first configuration depicted in Figure 15. Suppose first that x is not adjacent to 3. Color the ertices 1,..., 4 and y ith fie mutually distinct colors and the ertex x ith the color of 3. The precoloring can be extended to each shaded area by Proposition 11. We proceed similarly if x is not adjacent to 4 or y is not adjacent to 7 and 8. Hence, e can assume that all the four dashed edges depicted in Figure 15 are present in G. Since G is proper and Eulerian, it must be isomorphic to the triangulation T D. We no analyze the second configuration. Assume that the cycle x y contains a chord. Unless the subgraph of G induced by the ertices 1,..., 4, x and y is complete, the ertices 1,..., 4, x, y can be properly colored ith fie colors and the coloring can be extended to the shaded areas by Proposition 11. If the subgraph induced by the ertices 1,..., 4, x, y is complete, then the ertex x is adjacent to the ertices 4 = 7 and 6 and the ertex y to the ertices 6 and 7. It is easy to conclude that G must be the triangulation T C in such a case. Next, e assume that the cycle x y is chordless. If there is no ertex 23

24 of degree four inside the cycle x y, then by Theorem 15, the interior of the cycle x y contains a single ertex of degree six adjacent to all the ertices on the boundary of the cycle and the degree of y in G is fie (recall that G is proper) hich contradicts our assumption that G is Eulerian. Otherise, there is a non-contractible ertex inside the cycle x y. Since is non-contractible, all its four neighbors are among the ertices x, y, 5,..., 8. At least to such neighbors are not consecutie on the cycle x y and thus the cycle contains a chord since G is a triangulation. This iolates our assumption that the cycle is chordless. Let us no proceed ith the third configuration depicted in Figure 15. If y is not adjacent to 1, color the ertices 1,..., 8 ith four distinct colors, the ertex y ith the color of 1 = 5 and x ith the fifth color. The obtained coloring can be extended to each shaded area by Proposition 11. Next, it can be extended to and and e obtain a coloring of G ith 5 colors. An analogous argument applies if y is not adjacent to 7 or x is not adjacent to 5. Let us next assume that G contains all the edges y 1, y 7 and x 5. Let G be the near-triangulation bounded by the 4-cycle x 6 7 y. Since G is Eulerian and proper, the degree of y in G is een. So is the degree of 6 by Proposition 9. Hoeer, the degree of 2 = 6 in G is then odd hich is excluded by our assumption that G is Eulerian. The last configuration depicted in Figure 15 is the easiest to analyze. First, color the ertices 1,..., 8 ith four distinct colors and assign the ertex y the fifth color. Next, color the ertex x ith the color assigned to the ertex 3 = 8. The precoloring can be extended to each shaded area by Proposition 11 and it can eentually be extended to and. The final case is hen the triangulation ith the resistant configuration contains a non-contractible ertex ith three neighbors in common ith. Lemma 26. Let G be a proper Eulerian triangulation of the Klein bottle ith the resistant configuration and no contractible ertices. If G contains a noncontractible ertex that is adjacent to three of the ertices 1,..., 8, then G is either 5-colorable or it is the Eulerian triangulation T E depicted in Figure 16. Proof. By Lemma 24, unless G is 5-colorable, is adjacent to ertices i, i+1, i+2. By symmetry, e can assume that is adjacent to either 1, 2 and 3, or to 3, 4 and 5, see Figure 17. Let x be the neighbor of that is different from the ertices 1,..., 8. Let us first consider the case that is adjacent to 1, 2 and 3. Since is non-contractible, x must also be adjacent to 6. Let G 1 and G 2 be the subgraphs of G as depicted in Figure 17. By Proposition 11, there exist a coloring of G 1 such that the ertices x and 3,..., 6 are colored ith mutually distinct colors and a coloring of G 2 such that the ertices x and 6,..., 1 are colored ith mutually 24