An optimal chromatic bound for ($P_2+P_3$, gem)-free graphs
Arnab Char, T. Karthick
TL;DR
This work determines the smallest χ-binding function φ for the class of ($P_2+P_3$, gem)-free graphs. It proves χ(G) ≤ φ(ω(G)) with φ(1)=1, φ(2)=4, φ(3)=6, and φ(x)=\left\lceil \frac{5x-1}{4} \right\rceil for x≥4, and shows these bounds are tight via explicit extremal constructions. The approach combines Lovász’s blowup/perfection framework with a detailed C5-centered decomposition and a staged exclusion of small induced subgraphs (F1, F2, F3) to derive χ-coloring bounds that are optimal. The results advance χ-boundedness theory for natural graph classes by providing an exact, non-linear χ-binding function and a complete characterization of extremal graphs achieving equality. Overall, the paper delivers both a tight theoretical bound and a structural coloring methodology for this graph family.
Abstract
Given a graph $G$, the parameters $χ(G)$ and $ω(G)$ respectively denote the chromatic number and the clique number of $G$. A function $f : \mathbb{N} \rightarrow \mathbb{N}$ such that $f(1) = 1$ and $f(x) \geq x$, for all $x \in \mathbb{N}$ is called a $χ$-binding function for the given class of graphs $\cal{G}$ if every $G \in \cal{G}$ satisfies $χ(G) \leq f(ω(G))$, and the \emph{smallest $χ$-binding function} $f^*$ for $\cal{G}$ is defined as $f^*(x) := \max\{χ(G)\mid G\in {\cal G} \mbox{ and } ω(G)=x\}$. In general, the problem of obtaining the smallest $χ$-binding function for the given class of graphs seems to be extremely hard, and only a few classes of graphs are studied in this direction. In this paper, we study the class of ($P_2+ P_3$, gem)-free graphs, and prove that the function $φ:\mathbb{N}\rightarrow \mathbb{N}$ defined by $φ(1)=1$, $φ(2)=4$, $φ(3)=6$ and $φ(x)=\left\lceil\frac{1}{4}(5x-1)\right\rceil$, for $x\geq 4$ is the smallest $χ$-binding function for the class of ($P_2+ P_3$, gem)-free graphs.