On Color Isomorphic Pairs in Proper Edge Colourings of Complete Graphs

TL;DR

This paper investigates color isomorphic pairs in proper edge colorings of complete graphs, providing new upper and lower bounds for specific graph families using probabilistic methods.

Contribution

It introduces bounds for color isomorphism problems in complete graphs for certain graph families, extending previous work with new probabilistic techniques.

Findings

Upper bounds for $f_2(n,H)$ for certain rooted power graphs.

Matching lower bounds for 1-subdivisions of complete bipartite graphs.

Application of the random polynomial method of Bukh.

Abstract

Following the recent paper which initiated the study of colour isomorphism problems for complete graphs, we obtain upper bounds for $f_2(n,H)$ for a family of graphs $H$ obtained as the $K_0$-th rooted power of a balanced rooted tree for some sufficiently large $K_0$. The proof uses the random polynomial method of Bukh. We also obtain matching lower bounds for $1$-subdivisions of the complete bipartite graph.