On color isomorphic subdivisions

TL;DR

This paper investigates the minimum number of colors needed in a proper edge-coloring of complete graphs to avoid certain subgraph configurations, providing a lower bound for specific subdivisions of complete graphs.

Contribution

It establishes a new lower bound on the number of colors required to avoid two vertex-disjoint color isomorphic subdivisions of complete graphs, answering an open question.

Findings

Proves that f_2(n, H_t) = Ω(n^{1 + 1/(2t-3)}) for the 1-subdivision H_t of K_t.

Answers an open question posed by Conlon and Tyomkyn.

Provides insight into the coloring complexity of subdivided complete graphs.

Abstract

Given a graph $H$ and an integer $k\geqslant 2$, let $f_{k}(n,H)$ be the smallest number of colors $C$ such that there exists a proper edge-coloring of the complete graph $K_{n}$ with $C$ colors containing no $k$ vertex-disjoint color isomorphic copies of $H$. In this paper, we prove that $f_{2}(n,H_{t})=\Omega(n^{1+\frac{1}{2t-3}})$ where $H_{t}$ is the $1$-subdivision of the complete graph $K_{t}$. This answers a question of Conlon and Tyomkyn (arXiv: 2002.00921).