On graphs with no induced five-vertex path or paraglider

TL;DR

This paper investigates the structure of graphs that do not contain an induced path of five vertices or a paraglider, establishing a tight upper bound on their chromatic number relative to their clique number, and characterizing extremal cases.

Contribution

It provides a tight upper bound on the chromatic number for ($P_5$, paraglider)-free graphs and characterizes all graphs exceeding this bound, demonstrating the bound's optimality.

Findings

The chromatic number is at most 1.5 times the clique number for these graphs.

The bound is tight, as shown by the complement of the Clebsch graph.

No polynomial-time algorithm can approximate the chromatic number within a factor better than 1.5.

Abstract

Given two graphs $H_1$ and $H_2$, a graph is $(H_1,\,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ or $H_2$. For a positive integer $t$, $P_t$ is the chordless path on $t$ vertices. A paraglider is the graph that consists of a chorless cycle $C_4$ plus a vertex adjacent to three vertices of the $C_4$. In this paper, we study the structure of ($P_5$, paraglider)-free graphs, and show that every such graph $G$ satisfies $\chi(G)\le \lceil \frac{3}{2}\omega(G) \rceil$, where $\chi(G)$ and $\omega(G)$ are the chromatic number and clique number of $G$, respectively. Our bound is attained by the complement of the Clebsch graph on 16 vertices. More strongly, we completely characterize all the ($P_5$, paraglider)-free graphs $G$ that satisfies $\chi(G)> \frac{3}{2}\omega(G)$. We also construct an infinite family of ($P_5$, paraglider)-free graphs such that every graph $G$ in the family has $\chi(G)=\lceil \frac{3}{2}\omega(G) \rceil-1$. This shows that our upper bound is optimal up to an additive constant and that there is no $(\frac{3}{2}-\epsilon)$-approximation algorithm to the chromatic number of ($P_5$, paraglider)-free graphs for any $\epsilon>0$.