Let G be a simple graph on n vertices. LetH be either the complete graph Km or the complete bipartite graph Kr,s on a subset of the vertices in G. We show that G contains H as a subgraph if and only if βi,α(H) ≤ βi,α(G) for all i ≥ 0 and α ∈ Z. In fact, it suffices to consider only the first syzygy module. In particular, we prove that β1,α(H) ≤ β1,α(G) for all α ∈ Z if and only if G contains a ...

Polar and monopolar graphs are natural generalizations of bipartite or split graphs. A graph G = (V,E) is polar if its vertex set admits a partition V = A∪B such thatA induces a complete multipartite and B the complement of a complete multipartite graph. If A is even a stable set then G is called monopolar. Recognizing general polar or monopolar graphs is NP-complete and, as yet, efficient reco...

A tournament is an orientation of a complete graph and a multipartite or c-partite tournament is an orientation of a complete c-partite graph. If D is a digraph, then let d + (x) be the outdgree and d ? (x) the indegree of the vertex x in D. The minimum (maximum) out-degree and the minimum (maximum) indegree of D are denoted by + ((+) and ? ((?), respectively. In addition, we deene = minf + ; ?...

Bialostocki, A. and P. Dierker, An isoperimetric lemma, Discrete Mathematics 94 (1991) 221-228. A continuous version of the following problem is solved: Let G be a multipartite graph with a given partition of its vertex set, A, U A, U U A,,,. Find the maximum possible number of edges in G such that G has no connected component with more then t vertices.

A graph is called supermagic if it admits a labelling of the edges by pairwise different consecutive positive integers such that the sum of the labels of the edges incident with a vertex is independent of the particular vertex. Some constructions of supermagic labellings of regular graphs are described. Supermagic regular complete multipartite graphs and supermagic cubes are characterized.

Abstract. In this paper, the exact formulae for the generalized degree distance, degree distance and reciprocal degree distance of strong product of a connected and the complete multipartite graph with partite sets of sizes m0, m1, . . . , mr−1 are obtained. Using the results obtained here, the formulae for the degree distance and reciprocal degree distance of the closed and open fence graphs a...

We describe a cryptanalysis of the GGH15 multilinear maps. Our attack breaks the multipartite key-agreement protocol in polynomial time by generating an equivalent user private key; it also applies to GGH15 with safeguards. We also describe attacks against variants of the GGH13 multilinear maps proposed by Halevi (ePrint 2015/866) aiming at supporting graph-induced constraints, as in GGH15.

The average connectivity of a digraph is the average, over all ordered pairs of vertices, of the maximum number of internally disjoint directed paths connecting these vertices. Among the results in this paper, we determine the minimum average connectivity among all orientations of the complete multipartite graph K n1 ,n2, ... ,nk and the maximum average connectivity when all partite sets have t...

The critical group of a connected graph is a finite abelian group, whose order is the number of spanning trees in the graph, and which is closely related to the graph Laplacian. Its group structure has been determined for relatively few classes of graphs, e.g. complete graphs, and complete bipartite graphs. For complete multipartite graphs Kn1,...,nk , we describe the critical group structure c...