Degree-doubling graph families

Let G be a family of n-vertex graphs of uniform degree 2 with the property that the union of any two member graphs has degree four. We determine the leading term in the asymptotics of the largest cardinality of such a family. Several analogous problems are discussed.Comment: 9 page

A new upper bound on the number of neighborly boxes in R^d

A new upper bound on the number of neighborly boxes in R^d is given. We apply a classical result of Kleitman on the maximum size of sets with a given diameter in discrete hypercubes. We also present results of some computational experiments and an emerging conjecture

Clique coverings and claw-free graphs

Let C be a clique covering for E(G) and let v be a vertex of G. The valency of vertex v (with respect to C), denoted by val(C) (v), is the number of cliques in C containing v. The local clique cover number of G, denoted by lcc(G), is defined as the smallest integer k, for which there exists a clique covering for E(G) such that val(C) (v) is at most k, for every vertex v is an element of V(G). In this paper, among other results, we prove that if G is a claw-free graph, then lcc(G) + chi(G) <= n + 1. (C) 2020 The Author(s). Published by Elsevier Ltd