Families of Linearly $χ$-bounded Graphs without Chair or its Induced Sub-graphs

Abstract

A hereditary class H of graphs is $χ$-bounded if there is a $χ$-binding function f such that for every $G$ in $H$, $χ(G)$ less than or equal to $f(ω(G))$. Here we prove that if a graph $G$ is free of 1. {Chair; P$_4$+K$_1$} or 2. {Chair; HVN}, then $χ(G)$ is linearly bounded by maximum clique size of G. We further prove that if $G$ is free of 3. {P$_4$+K$_1$; P$_3$ $\cup$ K$_1$} or 4. {P4+K1; K$_2$ $\cup$ 2K$_1$} or 5. {HVN; P$_3$ $\cup$ K$_1$} or 6. {HVN; K$_2$ $\cup$ 2K$_1$} or 7. {K$_{5-e}$; P$_3$ $\cup$ K$_1$} or 8. {K$_{5-e}$; K$_2$ $\cup$ 2K$_1$}, then there is a tight linear $χ$-bound for $G$.