A Generalization of $\chi$-Binding Functions

Abstract

An induced hereditary class of graphs $\mathcal {G}$ admits a $\chi$-binding function, if there exists a function $f : \mathbb {N}\rightarrow \mathbb {R^{+}}$ such that $\chi (G) \le f(\omega (G))$ for every $G \in \mathcal {G}$. In this paper, we introduce a generalized version of $\chi$-binding function called $(\chi , \alpha , \omega )$-binding function. An induced hereditary class of graphs $\mathcal {G}$ admits a $(\chi ,\alpha ,\omega )$-binding function, if there exists a function $f : \mathbb {N}\rightarrow \mathbb {R^{+}}$ such that $\chi (G) \le f(\alpha (G),\omega (G))$ for every $G \in \mathcal {G}$. We prove that the class of $\{C_4,K_1+2K_2\}$-free graphs admits a linear $(\chi ,\alpha ,\omega )$-binding function $f(\alpha , \omega )=2\alpha +\omega$ and an $(\omega +2\alpha )$-coloring of G can be found in $O(n^5)$ time, where $n=|V(G)|$. We note that there exists no function $f : \mathbb {N}\rightarrow \mathbb {R^{+}}$ such that $\chi (G) \le f(\omega (G))$ or $\chi (G) \le f(\alpha (G))$ for every $\{C_4,K_1+2K_2\}$-free graph G. In addition, we prove that, for a general graph G with at least one vertex, $\chi (G)\le 2 ^ {\frac{\omega (\omega +1)}{2}} \alpha ^{\omega -1}$, where $\omega =\omega (G)$ and $\alpha =\alpha (G)$.