The Maximum Number of Cliques in a Graph Embedded in a Surface

k-Chromatic Number of Graphs on Surfaces

A well-known result (Heawood [6], Ringel [11], Ringel and Youngs [10]) states that the maximum chromatic number of a graph embedded in a given surface S coincides with the size of the largest clique that can be embedded in S, and that this number can be expressed as a simple formula in the Eulerian genus of S. We study maximum chromatic number of k edge-disjoint graphs embedded in a surface. We...

On the maximum number of cliques in a graph embedded in a surface

This paper studies the following question: given a surfaceΣ and an integer n, what is the maximum number of cliques in an n-vertex graph embeddable in Σ? We characterise the extremal graphs for this question, and prove that the answer is between 8(n−ω)+ 2 and 8n+ 2 2 ω + o(2), where ω is the maximum integer such that the complete graph Kω embeds inΣ . For the surfaces S0, S1, S2, N1, N2, N3 and...

The Maximum Number of Edges in a Strongly Multiplicative Graph

Cohen-Macaulay $r$-partite graphs with minimal clique cover

‎In this paper‎, ‎we give some necessary conditions for an $r$-partite graph such that the edge ring of the graph is Cohen-Macaulay‎. ‎It is proved that if there exists a cover of an $r$-partite Cohen-Macaulay graph by disjoint cliques of size $r$‎, ‎then such a cover is unique‎.

On the Maximum Number of Cliques in a Graph

A clique is a set of pairwise adjacent vertices in a graph. We determine the maximum number of cliques in a graph for various graph classes. For example, the maximum number of cliques in a planar graph with n vertices is 8(n − 2).