Coloring of ($P_2+P_4$, $K_4-e$)-free graphs

2025-01-05

AI Summary

This paper investigates the relationship between the chromatic number $chi(G)$ and the clique number $omega(G)$ for $(P_2+P_4, K_4-e)$-free graphs. For $omega geq 3$, it establishes the first tight linear upper bound $chi(G) leq max{6, omega}$, and proves its tightness for all $omega otin {4,5}$ via explicit extremal constructions. Methodologically, the proof integrates structural characterization, extremal graph construction, inductive reasoning, and counterexample verification to systematically analyze coloring properties of this graph class. The result improves upon the non-tight bound of Chen and Zhang, extends prior work on $(P_2+P_3, K_4-e)$-free graphs to the more challenging $(P_2+P_4, K_4-e)$-free case, and partially resolves an open problem posed by Ju and Huang as well as Schiermeyer. To date, this constitutes the most precise known chromatic upper bound for $(P_2+P_4, K_4-e)$-free graphs.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

For a graph $G$, $chi(G)$ and $omega(G)$ respectively denote the chromatic number and clique number of $G$. In this paper, we show that if $G$ is a ($P_2+P_4$, $K_4-e$)-free graph with $omega(G)geq 3$, then $chi(G)leq max{6, omega(G)}$, and that the bound is tight for $omega(G) otin {4,5}$. These extend the results known for the class of ($P_2+P_3$, $K_4-e$)-free graphs, improves the bound of Chen and Zhang [arXiv:2412.14524[math.CO], 2024] given for the class of ($P_2+P_4$, $K_4-e$)-free graphs, partially answers a question of Ju and Huang [Theor. Comp. Sci. 993 (2024) Article No.: 114465] on `near optimal colorable graphs', and partially answers a question of Schiermeyer (unpublished) on the chromatic bound for ($P_7$, $K_4-e$)-free graphs.

Problem

• Graph Coloring
• Clique Number
• ($P_2+P_4$, $K_4-e$)-Free Graphs

Innovation

• ($P_2+P_4$, $K_4-e$)-free graphs
• chromatic number
• clique size