‎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.

An [n, k, r]-partite graph is a graph whose vertex set, V , can be partitioned into n pairwisedisjoint independent sets, V1, . . . , Vn, each containing exactly k vertices, and the subgraph induced by Vi ∪ Vj contains exactly r independent edges, for 1 ≤ i < j ≤ n. An independent transversal in an [n, k, r]-partite graph is an independent set, T , consisting of n vertices, one from each Vi. An ...

An n-partite tournament is an orientation of a complete n-partite graph. An npartite tournament is a tournament, if it contains exactly one vertex in each partite set. Douglas, Proc. London Math. Soc. 21 (1970) 716-730, obtained a characterization of strongly connected tournaments with exactly one Hamilton cycle (i.e., n-cycle). For n ≥, we characterize strongly connected n-partite tournaments ...

‎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.

We show that if r s 2; n > r8; and G is a graph of order n containing as many r-cliques as the r-partite Turán graph of order n; then G has more than nr 1=r2r+12 cliques sharing a common edge unless G is isomorphic to the r-partite Turán graph of order n. This structural result generalizes a previous result that has been useful in extremal graph theory.

LetKn1,n2,...,np denote the complete p-partite graph, p > 1, on n = n1+n2+ · · ·+np vertices and let n1 ≥ n2 ≥ · · · ≥ np > 0. We show that for a fixed value of n, both the spectral radius and the energy of complete p-partite graphs are minimal for complete split graph CS(n, p− 1) and are maximal for Turán graph T (n, p).

Let μ (G) be the largest eigenvalue of a graph G and Tr (n) be the r-partite Turán graph of order n. We prove that if G is a graph of order n with μ (G) > μ (Tr (n)) , then G contains various large supergraphs of the complete graph of order r + 1, e.g., the complete r-partite graph with all parts of size log n with an edge added to the first part. We also give corresponding stability results.

Let r ≥ 2 and write K+ r (s1, . . . , sr) for the complete r-partite graph with parts of size s1 ≥ 2, . . . , sr with an edge added to the first part. Letting tr (n) be the number of edges of the r-partite Turán graph of order n, we prove the following theorems: (A) For all r ≥ 2, sufficiently small ε > 0 and sufficiently large n, every graph of order n with tr (n) + 1 edges contains a K + r ( ...

An n-tournament is an orientation of a complete n-partite graph. It was proved by J.A. Bondy in 1976 that every strongly connected n-partite tournament has an n-cycle. We characterize strongly connected n-partite tournaments in which a longest cycle is of length n and, thus, settle a problem in L. Volkmann, Discrete Math. 245 (2002) 19-53.

A graph G is called a complete k-partite (k ≥ 2) graph if its vertices can be partitioned into k independent sets V1, . . . , Vk such that each vertex in Vi is adjacent to all the other vertices in Vj for 1 ≤ i < j ≤ k. A complete k-partite graph G is a complete balanced kpartite graph if |V1| = |V2| = · · · = |Vk|. An edge-coloring of a graph G with colors 1, . . . , t is an interval t-colorin...