For a connected graph G containing no bridges, let D(G) be the family of strong orientations of G; and for any D ∈ D(G), we denote by d(D) the diameter of D. The orientation number d (G) of G is defined by → d (G)=min{d(D) |D ∈ D(G)}. In this paper, we study the orientation numbers of a family of graphs, denoted by G(p, q;m), that are obtained from the disjoint union of two complete graphs Kp a...

In their seminal paper Erdős and Szemerédi formulated conjectures on the size of sumset and product set of integers. The strongest form of their conjecture is about sums and products along the edges of a graph. In this paper we show that this strong form of the Erdős-Szemerédi conjecture does not hold. We give upper and lower bounds on the cardinalities of sumsets, product sets and ratio sets a...

In this paper simple formulae are derived for calculating the number of spanning trees of different product graphs. The products considered in here consists of Cartesian, strong Cartesian, direct, Lexicographic and double graph. For this purpose, the Laplacian matrices of these product graphs are used. Form some of these products simple formulae are derived and whenever direct formulation was n...

We attach topological concepts to a simple graph by means of the simplicial complex of its complete subgraphs. Using Forman’s discrete Morse theory we show that the strong product of two graphs is homotopic to the topological product of the spaces of their complexes. As a consequence, we enlarge the class of clique divergent graphs known to be homotopy equivalent to all its iterated clique graphs.

We define a diamonded odd cycle to be an odd cycle C with exactly two chords and either a) C has length five and the two chords are non-crossing; or b) C has length greater than five and has chords (x,y) and (x,z) with (y,z) an edge of C and there exists a node w not on C adjacent to y and C, but not x. In this paper, we show that given a diamonded odd cycle-free graph G, G is perfect if and on...

In 1979, two constructions for making partitionable graphs were introduced in (by Chv2 atal et al. (Ann. Discrete Math. 21 (1984) 197)). The graphs produced by the second construction are called CGPW graphs. A near-factorization (A; B) of a %nite group is roughly speaking a non-trivial factorization of G minus one element into two subsets A and B. Every CGPW graph with n vertices turns out to b...

A graph is a split graph if its vertices can be partitioned into a clique and a stable set. A graph is a k-split graph if its vertices can be partitioned into k sets, each of which induces a split graph. We show that the strong perfect graph conjecture is true for 2-split graphs and we design a polynomial algorithm to recognize a perfect 2-split graph.

Dujmović, Joret, Micek, Morin, Ueckerdt and Wood [J. ACM 2020] proved that for every planar graph $G$ there is a $H$ with treewidth at most 8 path $P$ such $G\subseteq H\boxtimes P$. We improve this result by replacing "treewidth 8" "simple 6".