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

TL;DR

This paper explores the generalized $ heta$-number of graphs, analyzing its properties, bounds, and exact values for specific graph classes, advancing understanding of graph coloring and related problems.

Contribution

It introduces new properties, bounds, and closed-form expressions for the generalized $ heta$-number across various graph classes and graph operations.

Findings

The sequence $( heta_k(G))_k$ is increasing and bounded by the graph order.

Closed-form expressions for $ heta_k(G)$ on Kneser, cycle, strongly regular, and orthogonality graphs.

Bounds on the product and sum of $k$-multichromatic numbers for a graph and its complement.

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.