On the generalized $\vartheta$-number and related problems for highly symmetric graphs

Abstract

This paper is an in-depth analysis of the generalized ϑ-number of a graph. The generalized ϑ-number, ϑ k (G), serves as a bound for both the k-multichromatic number of a graph and the maximum k-colorable subgraph problem. We present various properties of ϑ k (G), such as that the series (ϑ k (G)) k is increasing and bounded above by the order of the graph G. We study ϑ k (G) when G is the graph strong, disjunction and Cartesian product of two graphs. We provide closed form expressions for the generalized ϑ-number on several classes of graphs including the Kneser graphs, cycle graphs, strongly regular graphs and orthogonality graphs. Our paper provides bounds on the product and sum of the k-multichromatic number of a graph and its complement graph, as well as lower bounds for the k-multichromatic number on several graph classes including the Hamming and Johnson graphs.