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Graph Saturation in Multipartite Graphs

Abstract

Let GG be a fixed graph and let F{\mathcal F} be a family of graphs. A subgraph JJ of GG is F{\mathcal F}-saturated if no member of F{\mathcal F} is a subgraph of JJ, but for any edge ee in E(G)βˆ’E(J)E(G)-E(J), some element of F{\mathcal F} is a subgraph of J+eJ+e. We let ex(F,G)\text{ex}({\mathcal F},G) and sat(F,G)\text{sat}({\mathcal F},G) denote the maximum and minimum size of an F{\mathcal F}-saturated subgraph of GG, respectively. If no element of F{\mathcal F} is a subgraph of GG, then sat(F,G)=ex(F,G)=∣E(G)∣\text{sat}({\mathcal F},G) = \text{ex}({\mathcal F}, G) = |E(G)|. In this paper, for kβ‰₯3k\ge 3 and nβ‰₯100n\ge 100 we determine sat(K3,Kkn)\text{sat}(K_3,K_k^n), where KknK_k^n is the complete balanced kk-partite graph with partite sets of size nn. We also give several families of constructions of KtK_t-saturated subgraphs of KknK_k^n for tβ‰₯4t\ge 4. Our results and constructions provide an informative contrast to recent results on the edge-density version of ex(Kt,Kkn)\text{ex}(K_t,K_k^n) from [A. Bondy, J. Shen, S. Thomass\'e, and C. Thomassen, Density conditions for triangles in multipartite graphs, Combinatorica 26 (2006), 121--131] and [F. Pfender, Complete subgraphs in multipartite graphs, Combinatorica 32 (2012), no. 4, 483--495].Comment: 16 pages, 4 figure

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