Odd clique minors and chromatic bounds of {3$K_1$, paraglider}-free graphs

Abstract

A paraglider, house, 4-wheel, is the graph that consists of a cycle $C_4$ plus an additional vertex adjacent to three vertices, two adjacent vertices, all the vertices of the $C_4$, respectively. For a graph $G$, let $χ(G)$, $ω(G)$ denote the chromatic number, the clique number of $G$, respectively. Gerards and Seymour from 1995 conjectured that every graph $G$ has an odd $K_{χ(G)}$ minor. In this paper, based on the description of graph structure, it is shown that every graph $G$ with independence number two satisfies the conjecture if one of the following is true: $χ(G) \leq 2ω(G)$ when $n $ is even, $χ(G) \leq 9ω(G)/5$ when $n$ is odd, $G$ is a quasi-line graph, $G$ is $H$-free for some induced subgraph $H$ of paraglider, house or $W_4$. Moreover, we derive an optimal linear $χ$-binding function for {3$K_1$, paraglider}-free graph $G$ that $χ(G)\leq \max\{ω(G)+3, 2ω(G)-2\}$, which improves the previous result, $χ(G)\leq 2ω(G)$, due to Choudum, Karthick and Shalu in 2008.