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Note on the Thickness and Arboricity of a Graph View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF COMBINATORIAL THEORY, Series B 52, 147-l 51 ( 1991) Note On the Thickness and Arboricity of a Graph ALICE M. DEAN Department of Mathematics and Computer Science, Skidmore College, Saratoga Springs, New York 12866 JOAN P. HUTCHINSON* Department of Mathematics, Macalester College, St. Paul, Minnesota 5510.5 EDWARD R. SCHEMERMAN? Department of Mathematical Sciences, The Johns Hopkins University, Baltimore, Maryland 21218 Communicated by the Editors Received November 23, 1988 We prove that the thickness and the arboricity of a graph with e edges are at most La + 3/2 J and r-1, respectively, and that the latter bound is best possible. 0 1991 Academic Press, Inc. The thickness of a graph G, O(G), is the minimum number of planar graphs into which the edges of G can be partitioned, and the arboricity, Y(G), is the minimum number of acyclic graphs into which the edges of G can be partitioned. Nash-Williams [9] has determined a precise formula for the arboricity of a graph; namely, Y(G) = max - eH 1 ’ InH- 1 * Research supported in part by N.S.F. Grant RI18901458 and by Smith College. + Research supported in part by the Office of Naval Research contract NOO014-85-K0622. 147 00958956/91 $3.00 Copyright 0 1991 by Academic Press, Inc. All rights of reproduction in any form reserved. 148 DEAN, HUTCHINSON, AND SCHEINERMAN where the maximum is taken over all nontrivial induced subgraphs H, and where eH and nH denote the number of edges and vertices of H, respec- tively. From this and Euler’s formula for planar graphs it follows that Y(G) < 38(G); clearly 0(G) < Y(G). First we derive the O(A) b ound on the thickness of a graph with e edges, and then, usin the Nash-Williams formula, we obtain the best possible bound of r /- e/Z] on the arboricity, exhibiting, for each e, a graph with e edges whose arboricity achieves this bound. It was previously known that Y(G) = O(J) e since in [S] it is shown that Y(G) 6 r( l/2) JG], where n is the number of vertices in the graph, and in [7] that Y(G) < L5/4 + (l/2) ,/m J. Both these bounds are achieved by the complete graphs; the latter bound is asymptotic to our bound, but infinitely often is larger by 1. We use definitions and basic facts from [2]. THEOREM 1. If G is a simple graph with e edges, then O(G) d L&K- WI* ProoJ: We use induction on 1P’/) + IEl. If 1V/I + I El = 1, then G = Ki and 8(K,) = 0. Suppose the theorem has been proved for all graphs with 1Vi + /El < n + e and let G be a graph with n vertices and e edges. First, suppose there is a vertex v with deg(v) 6 a. By induction, G - v has thickness at most La + 3/2 J. Let k = Lm + 3/2 J and decompose G - v into k planar graphs H,, . Hk. Since deg(v) < k, we add to each of the Hi, for 16 ib deg(v), the vertex v and the edge from v to its ith neighbor. Note that so modified, the Hi’s are planar graphs whose union is G. On the other hand, suppose there is no vertex with degree at most &?. In this case, 2e = c deg(v) >n @ YE V(G) and therefore n < 2fi. Since the thickness of K,, is at most L(n + 9)/6 J (see [l, 3, 8, 11, 12]), we have Since the thickness of the complete graph on n vertices is O(n), the bound of this result is of the right order, but we believe that the constants are not best possible. Note that O(K,) is approximately &@, but wL,2, n/2) = rn2/(8n - 16)] is approximately &@ (see [4]). We conjecture that 8(G) d m + 0( 1) for any graph G. THICKNESS AND ARBORICITY OF A GRAPH 149 The previous proof with “arboricity” replacing “thickness” and the fact that Y(K,) < [n/2] leads to Y(G) < r&l. However, we prove the following stronger and best possible result. THEOREM 2. If G is a simple graph with e edges, then Y(G) < rJe/2a and this bound is best possible. ProoJ: By Nash-Williams’ result [9], there is a subgraph G’ of G with n’ vertices and e’ edges, and Y(G)= Y(C)= r& 1. If e’ 6 (n’ - 1)*/2 then (e’/(n’ - l))* < e’/2, and If e’ > (n’ - 1)*/2, we have Y(G) = Y(G') < l-&t) = 11; , and we compute n’- 1 e’> (n’- 1)*/2* $>T J Next, we show that the inequality is best possible in the sense that among all graphs with e edges there is one for which Y= r&l. The construction is as follows: Let n be the integer satisfying SO put e = (‘;) + k with 0 < k < n. We make a graph G with n + 1 vertices. Form a complete graph on n of them and let the degree of the remaining vertex be k. We claim that Y= Y(G) = rm]. We have Y< r&?l so we work to show Y> rJe/21. By the Nash-Williams result, 150 DEAN, HUTCHINSON, AND SCHEINERMAN Notice that -=-(z)+k n--l+! n 2 n’ Now if k/n < l/2 we have that Ya rn/21 and and therefore On the other hand, if k/n > l/2 then Yb [(n + 1)/21 and and therefore If G is a triangle-free graph, we can modify the above proof and apply Turan’s theorem to bound its arboricity by (l/2) A. This bound is achieved by the complete bipartite graph K,,.. This also gives a slightly improved thickness bound for triangle-free graphs. The proof technique of Theorem 1 is a simplification of that used in [6] to show that the thickness and arboricity of a graph of genus g are O(h). This technique leads to a simpler proof of the result in [6] that e(G) < 6 + ,/m. The proof technique of Theorem 2 is derived from that of [lo] where the maximum interval number of a graph is compared with the genus of a graph. ACKNOWLEDGMENTS The authors thank a referee, whose comments simplified the proof of Theorem 2, and H. S. Wilf for several helpful comments on the manuscript. REFERENCES 1. V. B. ALEKSEEV AND V. S. GONCHAKOV, Thickness of arbitrary complete graphs, Mat. Sb. 101, No. 143 (1976), 212-230. 2. M. BEHZAD, G. CHARTRAND, AND L. LESNIAK-FOSTER, “Graphs and Digraphs,” Wadsworth, Belmont, CA, 1979. THICKNESS AND ARBORICITY OF A GRAPH 151 L. W. BEINEKE AND F. HARARY, The thickness of the complete graph, Cunad. J. Math. 21 (1969), 850-859. L. W. BEINEKE, F. HARARY, AND J. W. MOON, On the thickness of the complete bipartite graph, Proc. Cambridge Philos. Sot. 60 (1964), l-6. N. CHIBA AND T. NISHIZEKI, Arboricity and subgraph listing algorithms, SIAM J. Comput. 14 (1985), 210-223. A. M. DEAN AND J. P. HUTCHINSON, Relations among embedding parameters for graphs, in “Proceedings Sixth Int’l. Conf. on Graphs and Applications, Kalamazoo, MI, 1988,” to appear. 7. H. N. GABOW AND H. H. WESTERMAN, Forests, frames and games: Algorithms for matroid sums and applications, in “Proceedings 20th ACM Symp. on Theory of Computing, 1988,” pp. 407421. 8. J. MAYER, Decomposition de K,, en trois graphes planaires, J. Combin. Theory Ser. B 13 (1972), 71. 9. C. ST. J. NASH-WILLIAMS, Decomposition of finite graphs into forests, J. London Math. Sot. 39 (1964), 12. 10. E. R. SCHJZINERMAN, The maximum interval number of graphs with given genus, J. Graph Theory 11 (1987), 441-446. 11. J. M. VASAK, The thickness of the complete graph, Notices Amer. Math. Sot. 23 (1976), A-479. 12. A. T. WHITE AND L. W. BEINEKE, Topological graph theory, in “Selected Topics in Graph Theory” (L. W. Beineke and R. J. Wilson, Eds.), pp. 1549, Academic Press, Orlando, FL, 1978. .
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