Equitable tree colouring of graphs
Yuping Gao, Allan Lo, Songling Shan

TL;DR
This paper extends equitable colouring concepts to tree and degenerate colourings, proving new bounds for graphs with bounded maximum degree and large size, confirming a conjecture for even maximum degree.
Contribution
It establishes new bounds for equitable tree colourings in graphs with maximum degree , confirming a conjecture for large graphs and even .
Findings
Every large graph with maximum degree has an equitable tree -colouring if (+2)/2.
Confirms a conjecture of Wu, Zhang, and Li for large graphs when is even.
Provides bounds for d-degenerate colourings in graphs with bounded maximum degree.
Abstract
Let and let be a simple graph with maximum degree . A -colouring of is an assignment of colours from to the vertices of . We call proper if adjacent vertices receive distinct colours, and equitable if the sizes of any two colour classes differ by at most one. The celebrated Hajnal--Szemer\'{e}di theorem states that a proper equitable -colouring exists whenever . In this paper, we study its tree colouring variant in which each colour class induces a forest. This is closely related to the vertex arboricity which was introduced by Chartrand, Kronk, and Wall. More precisely, we prove that if and , then every -vertex graph with maximum degree at most contains an equitable tree -colouring. This confirms a conjecture of Wu, Zhang, and Li…
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