Alpha labelings of straight simple polyominal caterpillars

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1 Apha abeings of straight simpe poyomina caterpiars Daibor Froncek, O Nei Kingston, Kye Vezina Department of Mathematics and Statistics University of Minnesota Duuth University Drive Duuth, MN 82-3, U.S.A. e-mai: {daibor,kings22,vezin9}@d.umn.edu Abstract Barrientos and Minion [] introduced the notion of snake poyomino graphs and proved that they admit an apha abeing. We introduce a reated famiy of graphs caed straight simpe poyomina caterpiars and prove that they aso admit an apha abeing. This impies that every straight simpe poyomina caterpiar with n edges decomposes the compete graph K 2kn+ for any positive integer k. Keywords: Apha abeing, gracefu abeing, graph decomposition. 2 Mathematics Subject Cassification: C8 Introduction In her tak at the Forty-Fifth Southeastern Internationa Conference on Combinatorics, Graph Theory, and Computing in Boca Raton in March, 2, Sarah Minion (an undergraduate student) presented her joint resuts with Christian Barrientos on apha abeings of snake poyominoes and other reated graphs []. The first author of this paper then proposed a generaization of their resut as a topic for the fina project in his graph theory cass. The two other co-authors (aso undergraduate students) wrote a project report that ater deveoped into the presented paper. Barrientos and Minion [] define a snake poyomino as a chain of m edge-amagamated four-cyces C, C 2,..., C m with the property that C shares one edge with C 2, C m shares one edge with C m, and for i = 2, 3,..., m, each C i shares one edge with C i and another edge with C i+. Note that no edge appears in more than two of those four-cyces. We generaize this notion and define a straight simpe poyomina caterpiar as foows. The spine of the caterpiar is a straight snake poyomino

2 Figure : Straight simpe poyomina caterpiar in which the edges of C i shared with C i and C i+ are non-adjacent, which means that every vertex is of degree at most three. The spine can be aso viewed as the Cartesian product P m+ P 2. We denote the vertices of the two paths as x, x,..., x m and y, y,..., y m, respectivey. A straight simpe poyomina caterpiar than can be constructed by amagamating at most one four-cyce to each of the edges x j x j+ and y y + for j, {,,..., m }. Notice that we can amagamate the four-cyces to none, one, or both of the two edges x j x j+ and y j y j+ for any admissibe vaue of j. The number of four-cyces in the spine wi be referred to as the ength of the caterpiar. An exampe is shown in Figure. A. Rosa [3] introduced in 9 certain types of vertex abeings as important toos for decompositions of compete graphs K 2n+ into graphs with n edges. A abeing of a graph G with n edges is an injection from V (G), the vertex set of G, into a subset S of the set {,, 2,..., 2n} of eements of the additive group Z 2n+. Let ρ be the injection. The ength of an edge xy is defined as (x, y) = min{ρ(x) ρ(y), ρ(y) ρ(x))}. The subtraction is performed in Z 2n+ and hence < (x, y) n. If the set of a engths of the n edges is equa to {, 2,..., n} and S {,,..., 2n}, then ρ is a rosy abeing (caed originay ρ-vauation by A. Rosa); if S {,,..., n} instead, then ρ is a gracefu abeing (caed β-vauation by A. Rosa). A graph admitting a gracefu abeing is caed a gracefu graph. A gracefu abeing ρ is said to be an α-abeing if there exists a number λ (caed the boundary vaue) with the property that for every edge xy G with ρ(x) < ρ(y) it hods that ρ(x) λ < ρ(y). Obviousy, G must be bipartite to aow an α-abeing. A graph admitting an α-abeing is caed an α- graph. For an exhaustive survey of graph abeings, see [2] by J. Gaian. Let G be a graph with at most n vertices. We say that the compete graph K n has a G-decomposition if there are subgraphs G, G, G 2,..., G s of K n, a isomorphic to G, such that each edge of K n beongs to exacty one G i. Such a decomposition is caed cycic if there exists a graph isomorphism φ such that φ(g i ) = G i+ for i =,,..., s and φ(g s ) = G. 2

3 Each gracefu abeing is of course aso a rosy abeing. The foowing theorem was proved by A. Rosa in [3]. Theorem.. A cycic G-decomposition of K 2n+ for a graph G with n edges exists if and ony if G has a rosy abeing. The main idea of the proof is the foowing. K 2n+ has exactn + edges of ength i for every i =, 2,..., n and each copy of G contains exacty one edge of each ength. The cycic decomposition is constructed by taking a abeed copy of G, say G, and then adding an eement i Z 2n+ to the abe of each vertex of G to obtain a copy G i for i =, 2,..., 2n. For graphs with an α-abeing, even stronger resut was proved by A. Rosa. Theorem.2. If a graph G with n edges has an α-abeing, then there exists a G-decomposition of K 2kn+ for any positive integer k. The proof is based on the observation that if we increase a abes in the partite set with abes exceeding λ by t, then a edge engths wi aso stretch by t. Hence, we can take k copies of G and increase the abes in the upper partite set in the j-th copy by jn, where j =,,..., k. This way we obtain edge engths, 2,..., nk, each exacty once. 2 Amagamation of apha abeed graphs Barrientos and Minion [] made the foowing observation. Theorem 2.. If G of order v with n edges and G 2 of order v 2 with n 2 edges are two α-graphs with boundary vaues λ and λ 2, respectivey, then there exist their edge-amagamation Γ of order v + v 2 2 with n + n 2 edges that is aso an α-graph with boundary vaue λ = λ + λ 2. For i =, 2 et X i be the partite sets with the ower abes, that is, at most λ i, and Y i the sets with the upper abes. Ca the respective abeings f and f 2. Further, et e = x y be the ongest edge of G of ength n and e 2 = the shortest edge of G 2 of ength. Then indeed f (x ) =, f (y ) = n, f 2 ( ) = λ 2, and f 2 ( ) = λ 2 +. Barrientos and Minion observed that one can amagamate x with and y with and increase the abes in X and Y by λ 2 and abes in Y 2 by n to obtain the desired graph Γ. The amagamated edge arising from e and e 2 is caed the ink. Notice that the shortest edge of Γ is in the subgraph arising from G whie the ongest one is in the subgraph arising from G 2. The edge-amagamation of G and G 2 as described above wi be denoted as G G 2. It is easy to observe that this concept can be used for consecutive amagamation of any number of α-graphs into a arger α-graph. We wi use that observation in the next section. 3

4 3 Construction Using the above resut, we now prove that every straight simpe poyomina caterpiar is an α-graph. The proof wi be performed by strong induction. Therefore, we wi first need to show suitabe α-abeings of certain base cases of our caterpiars. We define graphs O, B, I, J, U, and Y by their drawings in Figure 2. We aso find their α-abeings and present them in Figure 3. Notice that the edge abeed s in each graph in Figure 2 has ength induced by the abeing in Figure 3 and the edge abeed is the ongest one induced by the abeing in Figure 3. There is no edge abeed s in graph B since we never use it in the inductive step. The reason is that it does not admit the abeing required by Theorem 2.. x x s s y y x s y x s x y x x x x s s y y y y y Figure 2: Graphs Y, V, J, O, I, N, B (cockwise from upper eft) The assertion of the Lemma beow foows directy from the abeings in Figure 3. Lemma 3.. Graphs O, B, I, J, U, N and Y have α-abeings that induce ength on edge s = x y (except B) and ongest ength on edge = x y (graphs O, B, I) or = (graphs J, U, N, Y ). Moreover, vertices x and y are aways abeed so that f H (x ) = λ H and f H (y ) = λ H + for every graph H on the ist except graph B. We wi aso need the foowing Lemma, proved by Rosa [3]. Lemma 3.2. Let G with n edges admit an α-abeing f. Then the mapping f defined as f (x) = n f(x) is aso an α-abeing. Moreover, the ength of every edge is the same in both f and f.

5 Figure 3: Graphs Y, V, J, O, I, N, B with abeings We wi say that the abeing f is a dua abeing of f. One can aso observe that if the boundary vaues of f and f are λ and λ, respectivey, then λ = n λ. For any graph H of the ist above, we define its refection H as the graph horizontay symmetric with H. Athough they are mutuay isomorphic, we need to treat the graphs and their refections separatey since an amagamation of two caterpiars H and H 2 may give two non-isomorphic graphs. Lemma 3.3. A straight simpe poyomina caterpiars of ength m, where m = or m = 2, have an α-abeing f with the property that the ongest edge is x m y m with f(x m ) =. Proof. There are four caterpiars of ength one: O, I, B, and B. The former three have the abeing by Lemma 3. shown in Figure 3, the atter by Lemma 3.2. There are caterpiars of ength two. Four of them, J, U, N and Y have α-abeings as in Figure 3. Their refections J, U, N and Y have α-abeings by Lemma 3.2. The remaining ones are shown in Figure. Caterpiars D and X can be amagamated of two copies of O and I, respectivey, using Theorem 2.. Caterpiar T and its vertica refection can be obtained by amagamating O I and I O, respectivey. Caterpiar L arises from amagamation B O, and L foows by Lemma 3.2. Finay, V is amagamated as B I and V foows again by Lemma 3.2.

6 Figure : Graphs X, T, V, L, D (cockwise from upper eft) Now we are ready to prove our main resut. Theorem 3.. Every straight simpe poyomina caterpiar admits an α- abeing. Proof. We prove the caim by strong induction on the number of four-cyces in the spine. Let G m be a straight simpe poyomina caterpiar of ength m, that is, with m spine four-cyces. As before, we wi denote the spine path vertices by x, x,..., x m and y, y,..., y m, respectivey. The abeing wi be denoted by f. In fact, we wi be proving a stronger statement. Namey, that a such caterpiars admit an α-abeing such that x m y m is the ongest edge and f(x m ) =. It is easy to observe that if we have an α-abeing with f(x m ) > f(y m ) =, we can use the dua f instead to get f (x m ) =. Hence, we can assume that the condition f(x m ) = is aways satisfied. By Lemma 3.3, a caterpiars of engths one and two are α-graphs. Now for m 3 we ook at the subgraph of G m induced by the spine vertices x m, x m, y m, y m and possiby the vertices of the four-cyces amagamated to edges x m x m and y m y m, if there are any. If the subgraph is isomorphic to either I or O, then we create G m by removing a vertices of that subgraph except x m and y m. By the inductive hypothesis, G m has an α-abeing such that x m y m is the ongest edge and f(x m ) =.

7 We use Theorem 2. to amagamate G m with either I or O to obtain the desired α-abeing of G m with the ongest edge x m y m. We notice that f(x m ) > f(y m ) =, so we use the dua abeing f of G m and we are done. If the subgraph of G m induced by the spine vertices x m, x m, y m, y m and the vertices of the four-cyces amagamated to edges x m x m and y m y m is not isomorphic to either I or O, then it must be isomorphic to B. In this case we wi be removing vertices x m, x m, y m, y m, the neighbors of either x m and x m or y m and y m beonging to B and possiby the vertices of the four-cyces amagamated to edges x m 2 x m and y m 2 y m, if there are any. Then we receive G m 2 and a graph H induced by vertices x m 2, x m, x m, y m 2, y m, y m and the vertices of the cyces amagamated to them. Because we have excuded some cases above, H must be isomorphic to one of graphs J, U, N, Y or their horizonta refections J, U, N, Y. Now G m 2 has the required abeing by inductive hypothesis and H by Lemma 3.3. Hence, we amagamate edge x m 2 y m 2 with the shortest edge x y of H of ength one to obtain G m = G m 2 H which has the desired abeing by Theorem 2.. It is easy to check that the ongest edge is now x m y m and f(x m ) =. This competes the proof. The resut on decompositions of compete graphs into straight simpe poyomina caterpiars now foows directy from Theorems 3. and.2. Coroary 3.. Every straight simpe poyomina caterpiar with n edges decomposes the compete graph K 2kn+ for any positive integer k. Further research One can simiary define more genera poyomina caterpiars as foows. Reca that a poyomina snake P S m of ength m is a graph consisting of m edge-amagamated four-cyces C, C 2,..., C m with the property that C shares one edge with C 2, C m shares one edge with C m, and for i = 2, 3,..., m, each C i shares one edge with C i and another edge with C i+. No edge appears in more than two of those four-cyces. This definition is equivaent to that given by Barrientos and Minion in []. Obviousy, P S m is an outerpanar graph. We denote by x and y the ony two vertices of degree two in C and by x and y the ony two vertices of degree two in C m. A poyomina caterpiar P C m of ength m is a graph arising from P S m (caed spine) by an amagamation of any number of four-cyces to any of the outer edges of P S m except xy and x y. Such caterpiar is straight if the spine is the graph P m+ P 2, and is simpe if at any outer edge there is at most one amagamated four-cyce. The obvious next steps woud be to investigate α-abeings of simpe poyomina caterpiars, straight poyomina caterpiars, and genera poy-

8 omina caterpiars. Our guess is that they are isted here in the increasing order of difficuty. Even more genera case woud be if we aow amagamation of mutipe four-cyces at edges xy and x y. References [] C. Barrientos and S. Minion, Apha abeings of snake poyominoes and hexagona chains, manuscript. [2] J. Gaian, A Dynamic Survey of Graph Labeing, The Eectronic Journa of Combinatorics, (29), # DS. [3] A. Rosa, On certain vauations of the vertices of a graph, Theory of Graphs (Int. Symp. Rome 9), Gordon and Breach, Dunod, Paris (9),