2 1 GRAPHS WITH FORBIDDEN SUBGRAPHS

Transcription

1 Jnstitute for Social Res&yn^ 2 1 GRAPHS WITH FORBIDDEN SUBGRAPHS Gary Chartrand, Dennis Geller, and Stephen Hedetniemi Introduction. Many graphs which are encountered in the study of graph theory are characterized by a type of configuration or subgraph they possess. However, there are occasions when such graphs are more easily defined or described by the kinds of subgraphs they are not permitted to contain. For example, a tree can he defined as a connected graph which contains no cycles and Kuratowski [22] characterized planar graphs as those graphs which fail to contain subgraphs homeomorphic from the complete graph K q or the complete bipartite graph K_. The purpose of this article is to study, in a unified manner, several classes of graphs, which can be defined in terms of the kinds of subgraphs they do not contain, and to investigate related concepts. In ; the process of doing this, we show that many "apparently unrelated" results in the literature of graph theory are closely related. Several unsolved problems and conjectures are also presented. "^Research supported in part by a grant from the National Science Foundation (GN-25W+).

2 Definitions and Notation. Before beginning our study, we present definitions of a few basic terms and establish some of the notation which will be employed throughout the article. Those definitions not given here may be found in [l8]. The graphs under consideration here are the ordinary graphs, i.e., finite undirected graphs possessing no loops or multiple lines. The points of a graph G are usually denoted by u, v, w and the lines by x, y, z. If x joins the points u and v, then we write x = uv. The degree of a point u in a graph G is denoted deg u. The smallest degree among the points of G is denoted min deg G while the largest such number is max deg G. A subgraph H of a graph G consists of a subset of the point set of G and a subset of the line set of G which together form a graph. Two special but important types of subgraphs are the following. The subgraph induced by a set U of points of G has U for its point set and contains all lines of G incident with two points of U. The subgraph induced by a set Y of lines of G has Y for its line set and. contains all points incident with at least one line of Y. Two subgraphs are disjoint if they have no points in common and line-disjoint if they have no line in common. A connected component of a graph G is a maximal connected subgraph of G. A cutpoint of a connected graph G is a point whose removal disconnects G. A bridge is a line whose removal disconnects G. A block of G is a maximal connected subgraph of G containing no outpoints.

3 The connected components of G partition its point set while the blocks of G partition its line set. Two important classes of graphs are the complete graphs and the complete bipartite graphs. The complete graph K has each two of its p points adjacent. The complete bipartite graph n or K(m, n) has m + n points; and its point set can be partitioned into two subsets, one containing m points and the other n points, such that each point of one subset is adjacent with every point of the other subset but no two points in the same subset are adjacent with each other. In general, then, the complete n-partite graph K (P-[_5 P 2 > > P n ) n a s ^P^ points and its point set can be partitioned into subsets V^, 1 i n, such that v. = p. and two points u and v are adjacent if and only if u V. 1 1 j and v V,, where j ^ k. A subdivision of a graph H is a graph G^ obtained from H by replacing some line x = uv of H by a new point w together with the lines uw and vw. A graph G is then said to be homeomorphic from a graph H if G can be obtained from H by a finite sequence of subdivisions. Two graphs G^ and G^ are homeomorphic with each other if there exists a graph G- homeomorphic from both GL and G.

4 Graphs With Property P n. For a real number r, [r] and {r} denote the largest integer not exceeding r and the least integer not less than r, respectively. We say that a graph G has property P, where n is a positive integer, i f G contains no subgraph which is homeomorphic from the complete graph K n + 1 o r "the; complete bipartite graph K([ n * 2 ], ( n ^ 2 A few observations now follow readily from this definition. A totally disconnected graph is one which has no lines. A graph which contains no subgraph homeomorphic from or ^ necessarily has no lines. (The graph K is actually superfluous here since it is 1?^ homeomorphic from K^.) Thus, a graph has property P^ if and only i f i t is totally disconnected. A forest is a graph without cycles. If a graph contains no subgraph homeomorphic from K Q or K, i t is forbidden to contain cycles. (Here again ^ itself is homeomorphic from Hence the graphs with property P^ are the forests. An outerplanar graph is a graph G which can be embedded in the plane so that every point of G lies on the exterior region. Tang [27] has investigated several properties of outerplanar graphs, while Chartrand and Harary [13] have characterized outerplanar graphs as those graphs which fail to contain subgraphs homeomorphic from or K^, ^. Therefore, the graphs with property P^ are the outerplanar graphs. Whenever an outerplanar graph is encountered in this article, we shall assume it is embedded in the plane so that all of its points lie on the exterior region.

5 A planar graph is one which can be embedded in the plane. The well-known theorem of Kuratowski [22] states that a graph is planar if and only if it contains no subgraph homeomorphic from K or K ^. Hence, the graphs with property P^ are the planar, graphs. Throughout this article i t is.assumed all planar graphs are embedded in the plane. Thus far, no special name has been given to graphs having property P^, where n ^ 5. In this article, several results dealing with graphs having property P, 1 n k 9 are given. The similarity in these results lead to natural conjectures, which are also presented. It is no surprise that the definition of graphs with property P^ was inspired by Kuratowski's characterization of planar graphs. Another well-known characterization of planar graphs with an amazing.resemblance to Kuratowski's theorem involves contractions. One might well wonder i f the definition of property P^ could be given in an equivalent form using contractions. We now consider this question. A graph G' is said to be a contraction of a graph G if there exists a one-to-one correspondence between the point set of G' and the subsets determined by a partition of the point set of G such that each of these subsets induces a connected subgraph of G and such that two points of G' are adjacent if and only if there is a line joining points of the corresponding subsets. Let G' be a contraction of the graph G. The subgraph induced by a set of lines of G' is called a subcontraction of G. In [28] it is shown that any subcontraction of G can be realized by a contraction of a subgraph of G induced by a set of lines.

6 -6- It is known (see [19]) that homeomorphism is a special case of subcontraction, i.e., if a graph G contains a subgraph which is homeomorphic from a graph H, then H is a subcontraction of G. This, however, implies the following. Proposition 1. If a graph G has neither K n+1 nor K([ n * 2 ], [ n * 2 }) as a subcontraction, then G has property P. We now show that the converse of Proposition 1 holds for n = 1, 2,3, h. For n = k, we refer to a result in the literature. Theorem (Wagner [30], Harary and Tutte [19]) A graph is planar if and only if it has neither nor K as a subcontraction. For n = 1 the solution is immediate, for if G has or 2 as a subcontraction, then G has lines so that G has K as a 2 subgraph; thus G does not have property P.^.. We state this as: Proposition 2. A graph is totally disconnected if and only if it has neither nor ^ as a subcontraction. We now consider n = 2. Theorem 1. A graph is a forest if and only i f i t has neither nor 2 as a subcontration. Proof. By Proposition 1, it is sufficient to show that i f a graph G has either or ^ as a subcontraction, then G is not a forest.

7 Suppose that G has as a subcontraction. Let the points of he denoted by u^, u^, and u^ and the lines by x^ = u i u 2? 2 x = U2 U 3' B n^ x 3 = U1 U 3* ^^-^ion, denote the connected subgraph of G corresponding to u., i = 1, 2, 3, by G^. For each i = 1, 2, 3, the existence of the line x^ in the subcontraction of G implies the existence of a line y^ = v i v i+3 ^n ^ joining points in the appropriate subgraphs, where, say, v^, v^ G^, v^, G,-,, and V 3' Y{ 5 ^ ^3" ^ n c e ^s c o n n e c "ted, there exists a path P(v^, v^) joining and v^- all of whose points lie in G^. The path P(v^, v^) is trivial if v^ = v^. Corresponding statements can be made about G^ and G^ obtaining paths P(v 2, v^) and v ^)? a 1 1 o f whose points lie in G^ and G^, respectively. Thus G contains the cycle P( v - _3 y3 P^V3 5 5 V 5^5 y 2 5 P^v 23 v0> y l 5 V V t h e r e f o r e G i s n o t a forest. The proof in the case t h a t G has ^2 a s a subcontracti ion is entirely analogous. Theorem 2. A graph G is outerplanar i f and only if i t has neither nor 3 as a subcontraction Proof. Again, by Proposition 1, we need only show that if a graph G has either or ^ as a subcontraction, then G is not outerplanar. We show this for ^ only, the proof for being similar. Denote the points of the subcontraction K~ ~ by u., i = 1, 2, 3> ^> 1,3 1 and the lines by x x = and x 2 = U1 U V X 3 = U 1 U 5 5 *k = U 2 U 3' x 5 = x^ = u^u^. As before, we denote the connected subgraph of G

8 -8- corresponding to "by &, i = 1, 2, 3» ^ 5- Corresponding to each x.^ in ^ ^ there is a line y^ = v i w j_> where is a point of either G 1 or G-. As with the proof of Theorem 1, for i = 3, k, 5, there exists a path P(w. v w. _) joining points w. n and w. _ all of i-2' i+l y * i-2 I + I whose points belong to G^. The points v^, v^, lie in the connectei subgraph G, and hence there necessarily exists a point v 1 in G and three mutually disjoint paths in G 1 to the points v^, v^, and v^. (The point v' may coincide with some or all of v^, v^, and v^.) Denote the subgraph of Q induced by v' and the line sets of the three paths meeting at v' by H^. Similarly, there exists a subgraph H 2 of G^ consisting of a point v" and three mutually disjoint paths joining v" with v^, v<_, and v^. One now observes that the subgraph of G induced by the lines y and the lines in the paths P(w i 2, w i+]_) o r su ' D g ra Phs H 1 and is homeomorphic from ^ so that G is not outerplanar. We summarize our results below. Proposition 3- For n = 1, 2, 3, h, a graph G has property P i f n and only if i t has neither & n+1 nor K([ n * 2 ], { n * 2 }) as a subcontraction. The results of Proposition 3 cannot be extended beyond n = k. To see this, we illustrate the situation for n - 5- The graph G of Figure 1 has property P since i t has no points of degree 5 and only two points of degree k, thereby showing that G has no subgraph

9 homeomorphic from or from ^. The suhcontraction (indeed, contraction) determined by the sets, i =. 1, 2,..., 6, and {u, v} is the graph ^? however, w w G: w w w U V Figure 1 We now see that the concept of property has an equivalent formulation in terms of subcontraction only when 1 n k. This suggests the problem of investigating graphs with the property, say P^1, that they have neither K n + - ] _ nor K([ n g ~L a s a subcontraction. now made. A few general observations concerning graphs with property P^ are Proposition k. (i) If G is a graph with property P, then i t has property P for all m > n.

10 -10- (ii) If G and G^ are two graphs which are homeomorphic with each other, then G^ has property P^ if and only if G 2 has property P. n (iii) For every graph G, there exists a positive integer n such that G has property P. (iv) A graph G has property P n if and only i f every connected component of G has property P. (v) A graph G has property P^ if and only if every block of G has property P. (vi) If a graph G has property P, then every subgraph of G has property P. Of course, every totally disconnected graph has 0 lines. It is well known that the maximum number of lines in a forest with p points occurs when the forest is a tree and that this number is p - 1. Tang [27] showed that the maximum number of lines in an outerplanar graph with p points is 2p - 3? while it goes back to Euler's time that the maximum number of lines is a planar graph with p points is 3p - 6. In each case, minor restrictions on the size of p are necessary. This can be summarized as follows. Proposition 5. The maximum number of lines in a graph with p points and having property P IS (n - i)p - m, where p n and 1 n k. n One might very well conjecture that Proposition 5 can be extended so as to hold for all positive integers n. However, there is reason to

11 -11- believe that this result is valid for "small" values of n only. We state this as an open question. Problem 1. Determine all values of n for which the maximum number of lines in a graph with p points and having property is (n - l)p - where p n. A maximal graph having property P^ is a graph with p points having the maximum number of lines possible for such a' graph having p. points. By Proposition 5, then, a maximal graph having property P n has (n - l)p - l i n e s i f 1 ^ n <: If G is a graph with p points and property P^ and if G has p - 1 lines, then G is a tree and is therefore connected. This observation can be extended to graphs having property P^ or P^. A graph G is n-connected if the removal of any k points from G, 0 k < n, results in neither a disconnected graph nor the trivial graph consisting of a single point. Theorem 3* If G is a maximal graph having property P, 2 s n s i, with p points, p n, then G is (n - l) - connected. Proof. For n = 2, the result is immediate by our earlier remark since the 1-connected graphs are simply the connected graphs. For the case n = 3, let G be an outerplanar graph with p s 3 points and 2p - 3 lines. If G were not 2-connected, then G would contain a cutpoint v belonging to two blocks, say B, and B p,

12 -12- of G which are consecutive in the outerplanar embedding of G. Let x^ and x^ be consecutive lines (about v) belonging to B^ and B^, respectively. Suppose x^ = u-^v and x^ = U 2 V * I * ow l i n e u i u 2 c a n be -added so that in the resulting graph all points lie on the exterior region.. This implies that the resulting graph is outerplanar and contains more lines than G. This is a contradiction; thus G is 2-connected. Finally, consider a maximal planar graph with p ^ k points. If G were not 3-connected, then G would contain two points u and v whose removal disconnects G. Let C^ denote one component of the resulting disconnected graph, and let be the union of the other components. Since no point of is adjacent in G with a point of C 2, i t is possible to embed G in the plane so that u and v lie on the exterior region. We note that since G is maximal planar, u and v are necessarily adjacent. Clearly, the line uv. may be embedded so that i t does not lie on the exterior region of G. One now observes that a point of lying on the exterior region of G can be joined to a point of C^ on the exterior region in a planar manner. This, however, contradicts the fact that G is a maximal planar graph. Hence, G is 3-connected. Conjecture 1. If G is a maximal graph having property P^ and p points, where p ^ n 2> 2, then G is (n - l) - connected. Theorem 3 has the following corollary.

13 -13- Corollary 3a If G is a maximal graph having property P^ and p points, where p ^ n and 1 n s k, then min deg G n - 1. If Conjecture 1 is true, then, of course, Corollary 3a can be extended to include all positive integers n. Such an extension may be possible, however, without the validity of Conjecture 1. We state this as: Conjecture 2. If G is a maximal graph having property P^ and p points, where p ^ n, then min deg G ^ n - 1. We conclude this section with another result dealing with the degrees of maximal graphs having property P n - Theorem k. A maximal graph G having property P, 2 n h 3 with p points, p ^ n, has at least n points whose degrees do not exceed 2n - 3. Proof. Since G is a maximal graph having property P, 2 n <> k, i t has (n - l)p - ^) lines by Proposition 5. Therefore, the sum of the degrees of the points of G is (2n - 2)p - n(n - l ). Not all p points of G can have degree at least 2n - 2, for then the sum of the degrees of the points of G is at least (2n - 2)p. Hence, there must be enough points whose degrees are less than 2n - 2 so that the number (2n - 2)p is reduced by n(n - l ). By Corollary 3 a, however, every point of G has degree at least n - 1. Thus, the degree of no one point of G can reduce the number (2n - 2)p by more than n - 1. Therefore, there must be at least n points whose degrees do not exceed 2n - 3-

14 -lu- Since every graph having property P^ is contained in a maximal such one, we arrive at the following result. Corollary ha. A graph having property P, 2 ^ n S ^ with p points, p n, has at least n points whose degrees do not exceed 2n - 3 Corollary Ub. (i) Every forest with p ^ 2 points has at least two points of degree at most 1. (ii) Every outerplanar graph with p ^ 3 points has at least three points of degree at most 3- ( i i i ) Every planar graph with p k points has at least four points of degree at most 5 Recalling the proof of Theorem U, one would be led to believe that an extension of Corollary ha depends upon knowledge of the number of lines in a maximal graph having property P^ along with the validity of Conjecture 2. Since i t is felt that no maximal graph having property P g n has more than (n - D P - lines, we conjecture the following. Conjecture 3- A graph having property P^ with p points, p n, has at least n points whose degrees do not exceed 2n - 3*

15 -15- Line-Graphs and Total Graphs. The line-graph L(G) of a graph G which is not totally disconnected is the graph whose points can he put in one-to-one correspondence with the lines of G in such a way that two points of L(G) are adjacent if any only if the corresponding lines of G are adjacent. The total graph T(G) of a graph G is the graph whose points can he put in one-to-one correspondence with the set of points and lines of G in such a way that two points of T(G) are adjacent if any only if the corresponding elements of G are adjacent (if both elements are points or both are lines) or incident (if one element is a point and the other a line). The line-graph emenated from the work of Whitney [31] while the total graph originated with Behzad [1]. In this section, we investigate the relationship of line-graphs and total graphs with graphs having property P. If the line-graph L(G) of a graph G is to be a forest, then clearly G cannot contain a cycle nor can i t contain a point v such that deg v 3- This implies that each component of G must be a path. Another way of stating this which is convenient for our" purposes is the following. Proposition 6. The line-graph L(G) of a graph G is a forest if and only if max deg G 2 and if deg v = 2 for a point v of G, then v is a outpoint. It is interesting to compare this with the next theorem, due to Sedlacek [26].

16 -16- Theorem. The line-graph L(G) of a planar graph G is planar if and only if max deg G k and if deg v = k for a point v of G, then v is a outpoint. From these two results on forests and planar graphs, a conjecture on outerplanar graphs clearly arises which we present as our next theorem. Theorem 5- The line-graph L(G) of a graph G is outerplanar if and only if max deg G 3 and i f deg v = 3 for a point v of G, then v is a cutpoint. Proof. We make the initial observation that if a graph G satisfies the condition stated in the theorem, then every block of G is either a cycle or a single line and that no point belongs to more than one cycle. The blocks of L(G) therefore can arise from G in one of two ways: (l) from two adjacent bridges of G, and (2) a cycle C of G together with all the bridges of G which are incident with C. The blocks determined in (l) are themselves bridges. In (2), the line-graph of the cycle C is again a cycle C' having the same length as C. Suppose x is a bridge of G incident with C where x is adjacent with the lines y J and z of C. Let v, v, and v be the corresx J y' z ponding points in L(G), where then v and v lie on the cycle C. z y The presence of the line x in G produces in the block of L(G) the point v in L(G) v 7 along with the lines v v and v v but x x y x z no more. This block and hence all blocks of L(G) are outerplanar. Therefore, L(G) is outerplanar.

17 -17- Conversely, suppose the line-graph L(G) of a graph G is outerplanar. We must have max deg G ^ 3 for if max deg G S: h then there exists a point of G incident with four lines. These four lines, however, produce four mutually adjacent points in L(G), i.e., the graph K^, which contradicts the outerplanarity of L(G). Suppose, finally, there exists a point v of G such that deg v = 3 but that v is not a outpoint. It follows then that v lies on a cycle C whose lines are, say, x^, x^, x^., where v is incident with x^ and x^. Also, v must be incident with a line y which is a diagonal of C. (if y is not a diagonal, then it belongs to a path joining two points of C, but this is a subgraph homeomorphic from, and the line-graph of such a graph can easily be shown to contain a subgraph homeomorphic from as ^ 5 3 well). Suppose then y is also adjacent to x^ and x v j _ j be the point of L(G) which corresponds to x. for j = 1,. 2,..., k J and let u correspond to y. The line set {v.v j = 1, 2,..., k - 1} U {v-jv^, uv^, uv\ +1 } then belongs to a subgraph homeomorphic from ^, and this is a contradiction. One would now probably be led to conjecture that: The line-graph L(G) of a graph G with property P^ has property P^ if and only if max deg G n and if deg v = n for a point v of G, then v is a cutpoint. Unfortunately, this conjecture is not true, at least not true for all n, since, for example, i t is false when n = l6, and therefore probably false for n l6. We do, however, conjecture the validity of the statement for "small" n. We thus present the following question.

18 -18- Problem 2. Determine the values of n s> 2 for which the following statement is true: the line-graph L(G) of a graph G with property has property P n i f and only i f max deg G ^ n and if deg v = n for a point v of G, then v is a cutpoint. In [2], Behzad proved the following. Theorem- The total graph T(G) of a graph G is planar if and only if max deg G 3 and if deg v = 3 for a point v of G then v is a cutpoint Analogous to this result, we present the following. Theorem 6. The total graph T(G) of a graph G is outerplanar if and only i f max deg G 2 and i f deg v = 2 for a point v of G, then v is a cutpoint. Proof. We have already seen that if a graph G possesses the properties stated in the theorem, then each component of G is a path. It is only routine to show that T(G) is outerplanar. For the converse, we assume max deg G ^ 3 thereby implying the existence of a point in G incident with three lines. However, this point and the three lines correspond to four mutually adjacent points, i.e., K^. Thus, max deg G 2. If deg v = 2 for a non-cutpoint v of G, then v lies on a cycle C, where say w^ = v^v and w^ = v^v. The corresponding cycle C' in T(G) together with the path determined from v l 1 5 W J w2' V 2 P ro^- uces a subgraph homeomorphic from ^? which is a contradiction We now state our results in another way.

19 -19- Preposition 7- (i) The total graph T(G) of a graph G is planar if and only if its line-graph L(G) is outerplanar. (ii) The total graph T(G) of a graph G is outerplanar if and only if its line-graph is a forest. We conclude this section with a conjecture. Conjecture k. The total graph T(G) of a graph G has property p n +^ if and only i f its line-graph L(G) has property P, for n 2.

20 -20- The n-chromatic Number. In [12] the n-chromatic number of a graph G, denoted X^^O> w a s introduced and defined as the minimum number of colors needed in coloring the points of G so that no path of length n has all its points colored the same. This is equivalent to determining the fewest number of subsets into which the point set of G can be partitioned so that the subgraph induced by any subset contains no path of length n. The 1-chromatic number of a graph is then simply its chromatic number. It was proved in [12] that for every positive integer n there exists a planar graph G such that XqC^) = ^- If G is totally disconnected, then, of course, X^0- ) ~ 1 f o r a 1 1 n - Also, if G^ denotes a path of length n, then X n (&) = 2 s o that f r every positive integer n there exists a forest G such that X^^) =2. We now prove the 'expected' result for outerplanar graphs. By an (m, n)-coloring of a graph G, we mean a coloring of the points of G with m colors such that not all the points on any path of length n are colored the same. Theorem? For every positive integer n, there exists an outerplanar graph G such that Xj/^) = 3- Proof. The result is obvious for n = 1. Thus, let the positive in teger n s 2 be given. We now construct an outerplanar graph G such that xjg) = 3.

21 Consider the graph G^ which is constructed by adding an additional point to a path A of length n and joining this point to each point of A. (See Figure 2 for the case n = k). Clearly, G^ is outerplanar and X n (^i) = 2 - G r (n = h) Figure 2 In any (2, n)-coloring of the graph G 1, there exists a triangle whose points are colored with two colors, which we denote by 1 and 2. Furthermore, the triangle can be selected so that it contains a point colored 1, say w^, and a point colored 2, say w^, such that the line WjW 2 is on the exterior region of G 2> Denote the third point of this triangle by w. From G^ we construct a graph G 2 by adding to G^ a new point for each line lying on the exterior region of G.^ and joining each added point to the endpoints of the corresponding line. The graph G^ is outerplanar and contains 2(n + 2) points and has 2(n + 2) lines on its exterior region. (See Figure 3 for n = h). n Figure 3

22 -22- Any (2, n)-coloring of G 2 induces a (2, n)-coloring of G, which, as we have seen, produces a triangle ww^w2 3 a n d w i w 2 a ^ n e on the exterior region of G^. Since w^w^ is such a line, there is a point w^ in G^ hut not in G-^ which is adjacent to both w^ and w^. Whether w^ is colored 1 or colored 2, there is a triangle in G^ containing w^ whose points are colored with two colors. In addition, this triangle contains two adjacent points colored differently (w^ and w^ or w^ and w^) such that the line x joining them lies on the exterior region of 0. Finally, in G 2 there exist two paths P 0 and ~? 0 0 all of whose points are colored 1 and all of whose points are colored 2, respectively, such that the line x is incident with endpoints of both P and P p p and such that the sum of the Lyd dyd lengths of P 0 and V 0 0 is at least 2. Lyd We now construct an outerplanar graph G^ from G 2 by adding a point to G 2 for each line lying on the exterior region of G 2 and joining each such point to the two endpoints of the corresponding line of Gg. The graph G^ has k(n + 2) points and k(n + 2) lines on its exterior region. If X^^) ^ 2' then X^^) = 3 and we are finished. If, however, x^1^) = 2, then any (2, n)-coloring of G^ induces a (2, n)-coloring of G 2 and from our previous discussion, we see that any such coloring of G will produce in G- two paths P_ and 5 J i^o V 0 all of whose points are colored 1 and all of whose points are colored 2, respectively, such that the sum of the lengths of P and P p is at least 3-

23 -23- In a like manner, we construct the graphs G^, G,_... so that each is outerplanar. If for some i, X^G^) = 3 5 then, of course, we are finished; thus assume that for all i, X n (^ i) =2 so that any (2, n)-coloring of induces a (2, n)-coloring of G^. Inductively, we see that any (2, n)-coloring of G^ produces two paths P^ ^ and?2 ^ all of whose points are colored 1 and all of whose points are colored 2, respectively, such that the sum of the lengths of P. and V 0. I, i d., I is at least i. However, for G 0., either P. ^ -, or P 0 0 -, has 2n-l 5 l,2n-l 2,2n-l 5 length at least n contradicting the fact that G^n has a (2, n)- coloring. Therefore, X^^n-l^ * 3 > b u t s i n c e x n^ G i+l^ S ^ i ^ i ^ + 1 for all i, it follows that there exists an integer j s 2n - 1 such that y (G.) = 35 proving the theorem. We conclude this section with the following open question. Conjecture 5- For every two positive integers m and n, there exists a graph G with property P^ such that X m (^) = n.

24 The Partition Numbers n n and rr n '. As has already been noted, for every graph G there exists a positive integer n such that G has property P. Also, for a given graph G and positive integer m, it is clear that although G itself may not have property P, G has subgraphs which do have property P. This observation leads to the following problem. Given a graph G and a positive integer m, determine the fewest number of subgraphs into which G can be divided so that each subgraph has property P. There are two natural choices as to the type of subgraph to be considered, namely subgraphs determined by lines of G and subgraphs determined by points of G. We investigate both of these alternatives. The point-partition number ^(^Os n ^ 1, of a graph G is the minimum number of subsets into which the point set of G can be partitioned so that the subgraph induced by each subset has property P. Analogously, the line-partition number TT n '(G), n s 2, of G is defined as the minimum number of subsets into which the line set of G can be partitioned so that the subgraph induced by each subset has property P. The line-partition number TT! (G) is not defined for n = 1 since no graph containing lines can have property P^; on the other hand, TT^(G) is defined for a l l graphs G because a subgraph induced by a set of points can be totally disconnected. f The number Tr 2 (G) was introduced by Renyi and has been referred to as the arboricity of G. The arboricity of a graph has been studied by Nash-Williams [23] and Beineke [3].

25 -25- The minimum number of line-disjoint subgraphs into which a graph G can be divided so that each subgraph is outerplanar is the number rr^'(g), which we call the outerthickness of G. The line-partition number TT^'(G) originated with Tutte [29] and has been termed the thickness of G. The thickness of a graph has also been studied by Beineke and Harary [8, 9] n^ a Beineke, Harary, and Moon [10]. We observe that the point-partition number TT^(G) is the chromatic number of G. This famous concept has, of course, been the subject of numerous research articles during the past several decades. For convenience, we refer to the point-partition numbers ^ ( G ), TT^(G), and TT^(G) of a graph G as its point-arboricity, point-outerthickness, and point-thickness, respectively. Evidently, the partition numbers TT (G) and TT '(G) have not been n n studied for n ^ 5- A fundamental fact in the determination of the numbers TT (G) and TT! (G) is that we can restrict ourselves to conn^ ' n v ' nected graphs, indeed 2-connected graphs. Proposition 8. (i) The value of TT N (G) and TT ' (G) is the maximum of the values of these numbers on the components of G. (ii) The value of TT N (G) and TT 1 (G) is the maximum of the values of these numbers on the blocks of G. Since TT^(G) 1 and TT '(G) ^ 1 for a l l graphs G and for a l l defined values of n and since TT (G) TT (G) and TT '(G) TT 1 (G) for n m, we arrive at the following elementary observation which we state for later reference.

26 -26- PropQsition 9. If G Is a graph with property P^ and i f n : m, then TT (G) = 1 and TT ' (G) = 1. n n ' We now consider the point-partition numbers in more detail. As would probably be expected, for most graphs G and for small values of n, the numbers TT (G) are difficult to determine. An important class of graphs for which the numbers TT N (G) are easily determined is the family of complete graphs. This formula is derived by noting that a set of n points in a complete graph induces K, which has property P, but that a set of n + 1 points induces K, which does not have * n+1 5 property P. Proposition 10. For every two positive integers n and p, "^(^p) = {^} Since every graph with p points can be considered as a subgraph of K, we obtain an upper bound for T^C^O* Corollary 10a. For every graph G with p points and every positive integer n, TT (G) * n n In general, the upper bound given in Corollary 10a is not particularly good.. We now present a tighter upper bound along with a lower bound. For a graph G and for n s 1, denote by M n the maximum number of points in G which induce a subgraph having property P. The number M 1 is therefore the independence number of G, often denoted by 3 q (G).

27 -27- Theorem 8. Let G be a graph with p points and let n a 1. Then p - M _ <; n ( G ) ^ { -} + 1. M n n n J Proof. We first consider the lower bound. Let S, S 0,..., S be a minimum partition of the point set of G such that each induces a subgraph of G having property P n < Thus, k = T T n( G ) a n d S^ <. M N for i = 1, 2, k. Therefore, S.I k M = TT (G)«M, but S S. I = p so that TT (G) p/m. To establish the upper bound, let S be a set of points of G which induces a subgraph having property P n such that s = M n. If G - S denotes the subgraph of G obtained by removing the points in S (along with all incident lines), then it is clear that TT N (G - S) T T n( G ") " 1 - However, since G - S has p - M points, we can apply Corollary 10a n p - M and obtain the fact that TT (G - S) < f -}. Therefore, TT (G) n v L J ' n n v ' p - M The lower bound given in Theorem 8 serves as a generalization of the result on the chromatic number noted in Berge [11* p. 37] and Ore [2U, p. 255] while the upper bound generalizes the result in [20]. We have already seen that TT N (G) = 1 if G has property P^ and m ^ n. We now investigate the case m > n for certain ordered pairs (m, n). If (m, n) = (2, l ), then we are considering TT^(G), where G has property P, i.e., we are considering the chromatic number of a forest.

28 -28- This number, of course, is either 2 or 1 depending on whether'the forest has or f a i l s to have lines. Again, for later reference, we state this formally as : The (2, l) - Theorem. If G is a graph with property P, then TT-^G) 2, i.e., the chromatic number of a forest does not exceed 2. We next look into (m, n) = (3? 2), that is, the point-arboricity of an outerplanar graph. The (3, 2) - Theorem. If G is a graph with property P, then TT 2 (G) ^ 2, i.e., the point-arboricity of an outerplanar graph does not exceed 2. Proof. We proceed by- induction on the number p of points of G, the result being obvious for p = 1. Assume then for all outerplanar graphs H having p - 1 points, p ;> 2, that n 2 (H) 2. Let G be an outerplanar graph with p points. By Corollary Ub(ii), G contains a point u of degree 3 or less (in fact, at least three such points if p ^ 3)- The graph G - u is clearly outerplanar and since G - u contains p - 1 points, - u) 2. If TT 2 (G - u) = 1, then the points of G - u induce a forest as does {u] so TT 2 (G) 2; thus, we assume TT 2 (G - u) = 2. Let the points belonging to one forest of G - u be "colored" Q> and those of the second forest be colored (3. We show that we can color the point u either # or p so that the subgraph of G induced by the points colored a (respectively 0) is a forest.

29 If deg u < 3? then the problem is handled quite easily so, without loss of generality, we assume deg u = 3 and let v^,, and v^ be the points adjacent with u. If all three points are colored the same, say or, then by coloring u with 3, the subgraph of G induced by the points colored g is clearly a forest. If not all three points v^, v^, v^ are colored the same, then one of a and (3 is used only once in the coloring of these three points. We thus assume v^ is colored of while and v^ are colored p. We then color u with a- The subgraph F of G induced by the points colored a adds only the point u and the line uv^ to the forest induced by the points of G - u colored &. Therefore, F is necessarily a forest. This completes the proof. The technique used in the proof of the (3 5 2) - Theorem is essentially that employed by Kempe [21] in his famous false proof of the Four Color Conjecture. Another Kempian proof is used in our next theorem. The proof can also be given using a result of Tang [27]. The (3j l) - Theorem. If G is-a graph with property P^, then TT-^G) 3> i.e., the chromatic number of an outerplanar graph does not exceed 3. Proof. The- proof is by induction on the number p of points of G. The theorem is immediate, of course, for p = 1. Assume for all outerplanar graphs H having p - 1 points, p 2 2, that ^(H) ^ 3* Let G be an outerplanar graph with p points, which contains a point u such that deg u < 3* Since G - u is outerplanar and contains

30 -30- p - 1 points, TT- l (G - u) / 3 5 then hy coloring u with a color different from the two colors used for G - u, we see that a 3-coloring of G results which implies that n^(g) ^ 3- Therefore, we assume TT^(G - u) = 3 If deg u 2, then at most two colors are used to color the points adjacent with u. This leaves a color available with which to color u. The same conclusion can be arrived at if deg u = 3 and some color is used more than once in coloring the points adjacent with u. Hence, it remains only to investigate the case where deg u = 3 and all three colors are used for the points adjacent with u. Denote the points adjacent with u by v^, v^, and v^, where v^ is colored a., i = 1, 2, 3> and a. ^ a. for i ^ j. Consider the subi i j graph of G - u induced by all those points colored either or ot^. If this subgraph should happen to be disconnected and if v^ and v^ should happen to belong to different components of this subgraph, then by interchanging the colors and o/^ i n t n e component containing v^, we s t i l l have a proper 3-coloring of G - u in which the point v^ is now colored a^. We see now that we are able to color u with a^9 thereby producing a 3-coloring of G; thus rr^(g) 3- One now sees that one of the colors used in 3- c l r ing G - u can be made available for u implying that TT^(G) 3 unless possibly v^ and v. lie in the same component of the subgraph induced by points of G - u colored q>. or c*. for all i and j such that 1 5 i < j s 3, so we now assume this situation occurs. This implies that for 1 <. i < j ^3? there exists a path joining v. and v. all of whose points are

31 -31- colored o/. or a.. Any two of these paths have precisely one point i 3 in common (namely, v^, v^, or v^), for otherwise i t is easy to show that the graph G contains a subgraph homeomorphic from the complete "bipartite graph IC,, which is not possible since G is outerplanar. However, if G contains three such paths, one each between v. and v., 1 ^ i < j s 3, i J then G necessarily contains a subgraph homeomorphic from K^. Since this is not possible, we reach the conclusion that this situation cannot occur and thus TT^(G) 3- We now investigate the number TT N (G) for graphs G having property P^, i.e., for planar graphs. The (U, 3) - Theorem. If G is a graph with property P^, then YT^(G) 2, i.e., the point-outerthickness of a planar graph does not exceed 2. Proof. Let S^ denote the set of points of G which lie on the exterior region of G. Clearly, the subgraph of G induced by is outerplanar. Consider now the planar subgraph G - S^ of G. Let denote the set of points which lie on the exterior region of G - S^. Again, the subgraph of G induced by is outerplanar. If every point of G is in U S 2, then the result is obvious; if not, then let S^ denote the set of points of G which lie on the exterior region of G - (S^ U S^). No point of S^ is adjacent with a point of S^, for if u S.^, v and uv is a line of G, then v necessarily lies on the exterior region of G - S 1 implying that v G S g so that v S^. Hence, the subgraph

32 -32- of G induced by U S^ in the union of the subgraph induced by and the subgraph induced by S^. Since each of these subgraphs is outerplanar, the subgraph induced by S^ U is outerplanar. In a like manner, we define, if necessary, the subsets S^, S,_, etc. If we let = ^2K+1 a n d ^2 = ^2k 3 ^ n e n ^ a n ana l g us argument to that just given the subgraph induced by V^, i = 1, 2, is outerplanar so that TT 3 (G)> 2. We now return to a Kempian proof for our next result. The (k, 2) - Theorem. If G is a graph with property P^, then TT 2 (G) 3, i.e., the point-arboricity of a planar graph does not exceed 3- Proof. We employ an induction proof on the number p of points of G with the result following trivially for p = 1. Assume for all planar graphs H having p - 1 points, p 2: 2, that TT 2 (H) ^ 3- Let G be a planar graph with p points. By Corollary Ub(iii), G contains a point u of degree 5 or less (in fact, at least four such points if p ^ k). Since G - u is planar and has p - 1 points, TT 2 (G - u) 3- If TT 2 (G - u) <: 2, then let V- L and V 2 be subsets which partition the point set of G - u such that each subset induces a forest. Then in G each of V^, V 2, and [u] induces a forest implying that TT 2 (G) 3«We thus assume TT 2 (G - u) = 3- Let S^, S 2, and S^ constitute a partition of the point set of G - u so that each S. induces a forest. If some S. contains no point l I adjacent with u, say S^ is such a set, then S^ U {u} must induce a

33 -33- forest so that TTg(G) 3- On "the other hand, if every contains a point of G adjacent with u, then some set must contain precisely one point adjacent with u since deg u 5. Assume contains a point v adjacent with u but has no other point adjacent with u. The set U [u] induces a subgraph F which adds only the point u and the line uv to the subgraph induced by S^; thus F is a forest. Hence, each of S 1 U {u}, Sg, and induces a forest showing that TT 2 (G) <; 3- This brings us to the case (m, n) = (4, l ), for which, unfortunately, we cannot supply a proof of the desired result for this is the famous Four Color Conjecture. For a complete discussion of this problem see Ore [25]. We state this conjecture in our terminology. The (k, l) - Conjecture. If G is a graph with property P^, then TT-^G) 4, i.e., the chromatic number of a planar graph does not exceed k. With the" 1 " exception of (m, n) = (4, l ), i.e., with the exception of the Four Color Problem, we have shown that for all ordered pairs (m, n), with 1 n m h 3 the point-partition number TT (G), where G has property P, does not exceed m - n + 1. The Four Color Conjecture also m satisfies this inequality. I t therefore seems natural to make the f o l lowing a 11-encompassing conjecture, where m 2: n. The (in, n) - Conjecture. If G is a graph with property P, then TT (G) s-.m -. n + 1.

34 -3k- We now turn our attention to the line-partition numbers TT! (G). An apparently very tight lower bound is given by the following formula, Theorem 9- Let n ^ 2 and let G be a graph which is not totally disconnected having p points, where p ^ n. Then TT '(G) - max { Q" )> where q, is the maximum number of lines in any subgraph of G induced by k points and is the maximum number of lines in any graph with k points and having property P. Proof. The inequality stated in the theorem follows by simply observing that in any line-disjoint decomposition of G into subgraphs having property P, the set of q lines must be partitioned so that none of n K the aforementioned subgraphs contains more than of these lines. For n = 2, 3 3 and k 9 we have the three succeeding corollaries. Corollary 9a. If G is a graph with p points which is not totally disconnected, then q k TT 2 '(G) 2:.max { - }. 2 k p Corollary 9^> If G is a graph with p ^ 3 points which is not totally disconnected, then q k TT '(G) max { -=r = }. 3 ' 3

35 -35- Corollary 9c If G is a graph with p a= k points which is not totally disconnected, then q k TT u '(G) > max { -xr T }. Nash-Williams [23] showed that the inequality in Corollary 9a is actually an equality, i.e., the arboricity of a graph G is given by \ TT ' (G) = max { T }. 2 k 2*k<;p " 1 It is known that there exist graphs such that the inequality stated in Corollary 9c is strict, for example, the graphs and K^Q. However, no example is known for which TT^'(G) exceeds that lower bound by more than one. The situation for TT^'(G) is undetermined. Just as we did for the point-partition numbers, we investigate the line-partition numbers TT ' (G) f o r graphs G with property P^, where 2 n m k. By Proposition 9, TT 1 (G) = 1 when n = m. The first case we consider is m = 3 and n = 2, which we denote by [3? 2] to distinguish i t from (3j 2), the corresponding "point" problem. The [3, 2] - Theorem. If G is a graph with property P, then TT 2 '(G) 2, i.e., the arboricity of an outerplanar graph does not exceed 2. Proof. Since we wish to determine the arboricity of a graph, we can refer to Nash-Williams' formula. Since G is outerplanar, for all k such that 2 k p, where p is the number of points of G, we must have

36 -36- q 2k? k q. <: 2k - 3. Since [-rr 7} = 2, i t follows that [r 7} ^ 2 so that k K 1 K. 1 TTg^Gr) has the value 1 or 2. The next theorem can he proved in a completely analogous manner using the result of Nash-Williams. The [U, 2] - Theorem. If G is a graph with property P^, then TT 2 '(G) <: 3, i.e., the arboricity of a planar graph does not exceed 3- This brings us to the case [m, n] = [U, 3]- As with the final case we considered in the point-partition problem, we find ourselves unable to give an answer to the question. Since TT^'CG) ^'(G), i t follows by the [4, 2] - Theorem, that the outerthickness of a planar graph cannot exceed 3. However, we know of no planar graph G for which TT^'(G) = 3; thus, we strongly suspect the following conjecture to be true. The [4, 3] - Conjecture. If G is a graph with property P^, then n^'(g) 2, i.e., the outerthickness of a planar graph does not exceed 2. We conclude this section with a conjecture involving the line-partition numbers of graphs with property P. The [m, n] - Conjecture. If G is a graph with property P, then TT '(G) m - n + 1, where 2 n <: m.

37 -37- The Dual Numbers TT"^ and TT^1. We have seen that with every graph G there are associated the problems of determining the minimum number of elements in a partition of the point set or line set of G so that each resulting subset induces a subgraph of G having property P. In this section, we introduce numbers which are, in a sense, "dual" to the point-partition numbers TT n (G) and line-partition numbers TT n '(G). The dual point-partition number TT n (G), n 1, of a graph G is defined to be the maximum number of disjoint point-induced subgraphs contained in G so that each subgraph does not have property P. Similarly, the dual line-partition number "^'(co, n 2> 1, is the maximum number of line-disjoint subgraphs contained in G so that each subgraph fails to have property P. By Proposition U(vi), it is a simple observation that ~^n(^) - 0 and tr n f(g) =0 if and only i f the graph G has property P. Since every graph G not having property P^ requires at least n + 1 points and [ ] { l i n e s, n 3, we obtain the following bounds on the dual numbers. Proposition 11. If G is a graph with p points and q lines, then V G > * ^ ] and where A(l) = 1, A(2) = 3 3 and A(n) = [ n * 2 ] { n * 2 } for n a 3.

38 38- Because any subgraph containing lines clearly cannot have property P^, for any graph G, ^'(G) = q, the number of lines of G. Unlike the numbers TT (G) and TT '(G), most of the dual numbers are n n ' invariant under homeomorphism, as we shall now see. Theorem 1Q. If the graph G^ is homeomorphic with the graph G^, then TT N (G 1 ) = TT N (G 2 ) and (G^ = n^gg), for n ;> 2. Proof. We prove only the "point" half of the theorem, the "line" version having a similar proof. Since G^ is homeomorphic with G 2, there exists a graph G^ which can be obtained from each of G.^ and G 2 by a sequence of subdivisions. To show that TT N (G 1 ) = 1T (G 2 ) S n ^ 2, i t is clearly sufficient to prove that the value of the number rt^ for one of G^ and G 2, say G^, equals that for G^. To prove this, however, it is sufficient to show that rr (G N ) = "TT (G '), where G ' is a subdivision of G,. n K 1' n v 1 ' Thus, there is a line x = uv of G 1 which has been replaced by a new point w and the two new lines uw and wv to obtain G-^'. Assume TT (G,) = k. If k = 0, then G. has property J P as does n' r 1 * * n G^', by Proposition 4(ii); thus, "^(G^*) = 0 also. Therefore, we proceed under the supposition that k 1. Consider a set of k disjoint, point-induced subgraphs of G^ such that each subgraph does not have property P. If none of these subgraphs contain both the points u and v, then this collection of subgraphs is also a set of k disjoint, point-induced subgraphs of G^ so that n (G.^) k. If, on the other

39 -39- hand, one of the point-induced subgraphs, say H, contains both u and v, then.a point-induced subgraph H' of G^' which is homeomorphic to H can be produced by replacing the line uv of H by the point w and the lines uw and wv. The subgraph H' and the subgraphs of different from H then form a set of k disjoint, point-induced subgraphs of G^1. Hence, in this case also, TT (G^!) ^ k. We now show the assumption that * T T n (^1') > k leads to a contradiction. Suppose G-^' contains k + 1 disjoint, point-induced subgraphs, each failing to have property P. If none of these subgraphs contain the point w and none contain both point u and v, then these subgraphs form a set of k + 1 disjoint, point-induced subgraphs of G^ also, and this is a contradiction. If none of these subgraphs contain w but some subgraph S 1 contains both u and v, then a point-induced subgraph S of G^ is produced by adding the line uv to S'. Since S' has a subgraph homeomorphic with either K n+1 of K([ n * 2 ], [ ^ )) so must S, and again a contradiction arises. Suppose now that some subgraph T' in the set of k + 1 disjoint, point-induced subgraphs of G^' contains w. If T 1 also contains u and v, a subgraph T of G homeomorphic with T! is produced by deleting w from T 1 and inserting the line uv. Since T is pointinduced, we have a set of k + 1 disjoint, point-induced subgraphs of G^', each without property P, and again a contradiction. If T' contains at most one of u and v, then because n 2, the subgraph