Edges not covered by monochromatic bipartite graphs

TL;DR

This paper investigates the maximum edges not in monochromatic copies of acyclic graphs in edge colorings, proving new bounds and counterexamples to previous conjectures, especially for bipartite graphs and trees.

Contribution

It establishes exact values for certain acyclic graphs, provides tight bounds for bipartite graphs with tails, and offers counterexamples to existing conjectures about trees in edge colorings.

Findings

Proved $f(n,H)=ex(n,H)$ for spiders and double brooms.

Provided tight bounds for bipartite graphs with tails, answering Keevash and Sudakov's question negatively.

Showed bounds for $f_{2k}(n,P_{2k})$ are tight when $2k-1$ is prime, giving a negative answer to a conjecture.

Abstract

Let $f_k(n,H)$ denote the maximum number of edges not contained in any monochromatic copy of~$H$ in a $k$-coloring of the edges of $K_n$, and let $ex(n,H)$ denote the Tur\'an number of $H$. In place of $f_2(n,H)$ we simply write $f(n,H)$. Keevash and Sudakov proved that $f(n,H)=ex(n,H)$ if $H$ is an edge-critical graph or $C_4$ and asked if this equality holds for any graph $H$. All known exact values of this question require $H$ to contain at least one cycle. In this paper we focus on acyclic graphs and have the following results: (1) We prove $f(n,H)=ex(n,H)$ when $H$ is a spider or a double broom. (2) A \emph{tail} in $H$ is a path $P_3=v_0v_1v_2$ such that $v_2$ is only adjacent to $v_1$ and $v_1$ is only adjacent to $v_0,v_2$ in $H$. We obtain a tight upper bound for $f(n,H)$ when $H$ is a bipartite graph with a tail. This result provides the first bipartite graphs which answer the question of Keevash and Sudakov in the negative. (3) Liu, Pikhurko and Sharifzadeh asked if $f_k(n,T)=(k-1)ex(n,T)$ when $T$ is a tree. We provide an upper bound for $f_{2k}(n,P_{2k})$ and show it is tight when $2k-1$ is prime. This provides a negative answer to their question.