Improved approximative multicoloring of hexagonal graphs

TL;DR

This paper presents an improved approximation method for multicoloring hexagonal graphs, reducing the ratio from 4/3 to 5/4 and providing a more efficient coloring strategy.

Contribution

The authors develop a new multicoloring approach for hexagonal graphs that improves the asymptotic approximation ratio from 4/3 to 5/4.

Findings

Achieved a multicoloring with at most 15 ⌊ω(G)/12⌋ + 18 colors

Improved the asymptotic approximation ratio from 4/3 to 5/4

Provided a proper multicoloring method for weighted hexagonal graphs

Abstract

In 1999, McDiarmid and Reed conjectued that the approximation ratio $9/8$ of multichromatic number to weighted clique number asymptotically is the best possible for general weighted hexagonal graphs. We prove that there is a proper multicoloring of $G$ that uses at most $15 \lfloor\frac{\omega(G)}{12}\rfloor+ 18 $ colors improving the best previously known asimptotic ratio from 4/3 to 5/4.