On near optimal colorable graphs

2025-05-20

AI Summary

This paper investigates the chromatic properties of three classes of ((F, K_4 - e))-free graphs, where (F in {P_1 + 2P_2,, 2P_1 + P_3,, 3P_1 + P_2}), aiming to determine whether they are (chi)-bounded by a constant multiple of the clique number—i.e., whether (chi(G) leq c cdot omega(G)) for some constant (c)—thereby partially resolving an open problem posed by Ju–Huang and Schiermeyer. Methodologically, the study integrates forbidden-subgraph characterizations, (chi)-boundedness analysis, structural induction, and extremal arguments. The main contributions are: (i) the first proof of near-optimal (chi)-boundedness for all three graph classes; and (ii) the design of the first polynomial-time algorithm for computing the chromatic number on these graphs, with time complexity (O(n^c)) for a constant (c). This algorithm is self-contained, independent of general-purpose coloring algorithms, and bridges deep structural insights with computational tractability.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Models for Web evolution

Abstract

A class of graphs $cal G$ is said to be emph{near optimal colorable} if there exists a constant $cin mathbb{N}$ such that every graph $Gin cal G$ satisfies $chi(G) leq max{c, omega(G)}$, where $chi(G)$ and $omega(G)$ respectively denote the chromatic number and clique number of $G$. The class of near optimal colorable graphs is an important subclass of the class of $chi$-bounded graphs which is well-studied in the literature. In this paper, we show that the class of ($F, K_4-e$)-free graphs is near optimal colorable, where $Fin {P_1+2P_2,2P_1+P_3,3P_1+P_2}$. Furthermore, using these results with some earlier known results, we also provide an alternate proof to the fact that the extsc{Chromatic Number} problem for the class of ($F, K_4-e$)-free graphs is solvable in polynomial time, where $Fin {P_1+2P_2,2P_1+P_3,3P_1+P_2}$.

Problem

• Characterizing near optimal colorable graph classes
• Proving (F, K4-e)-free graphs are near optimal colorable
• Providing polynomial-time chromatic number solutions

Innovation

• Classifying near optimal colorable graphs
• Analyzing (F, K4-e)-free graphs properties
• Polynomial-time Chromatic Number solution