Weak degeneracy of regular graphs

TL;DR

This paper establishes the exact lower bound of weak degeneracy for d-regular graphs, disproving a previous conjecture and contributing to the understanding of graph coloring algorithms.

Contribution

It proves the tight lower bound of weak degeneracy for d-regular graphs, refuting Bernshteyn and Lee's conjecture.

Findings

Lower bound of weak degeneracy for d-regular graphs is exactly ⌊d/2⌋ + 1

The result is tight and refutes the previous conjecture

Advances understanding of graph degeneracy in relation to coloring algorithms

Abstract

Motivated by the study of greedy algorithms for graph coloring, Bernshteyn and Lee introduced a generalization of graph degeneracy, which is called weak degeneracy. In this paper, we show the lower bound of the weak degeneracy for $d$-regular graphs is exactly $\lfloor d/2\rfloor +1$, which is tight. This result refutes the conjecture of Bernshteyn and Lee on this lower bound.