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

Abstract

This paper is an in-depth analysis of the generalized $\vartheta$-number of a graph. The generalized $\vartheta$-number, ${\vartheta }_k(G)$, 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 ${\vartheta }_k(G)$, such as that the sequence $({\vartheta }_k(G){)}_k$ is increasing and bounded from above by the order of the graph $G$. We study ${\vartheta }_k(G)$ when $G$ is the strong, disjunction or Cartesian product of two graphs. We provide closed form expressions for the generalized $\vartheta$-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.