Singular Tur\'an numbers and WORM-colorings

TL;DR

This paper determines exact values of singular Turán numbers for certain small graphs and large enough parameters, and explores their connection to H-WORM colorings, advancing extremal graph theory understanding.

Contribution

It provides exact values of $T_S(n,K_3)$ for all relevant n and extends results to $T_S(n,K_{r+1})$, also linking singular Turán numbers to H-WORM colorings.

Findings

Exact values of $T_S(n,K_3)$ for all $n$ mod 4.

Determination of $T_S(n,K_{r+1})$ for large divisible n.

New bounds on edges in graphs with H-WORM colorings.

Abstract

A subgraph $H$ of $G$ is \textit{singular} if the vertices of $H$ either have the same degree in $G$ or have pairwise distinct degrees in $G$. The largest number of edges of a graph on $n$ vertices that does not contain a singular copy of $H$ is denoted by $T_S(n,H)$. Caro and Tuza [Theory and Applications of Graphs, 6 (2019), 1--32] obtained the asymptotics of $T_S(n,H)$ for every graph $H$, but determined the exact value of this function only in the case $H=K_3$ and $n\equiv 2$ (mod 4). We determine $T_S(n,K_3)$ for all $n\equiv 0$ (mod 4) and $n\equiv 1$ (mod 4), and also $T_S(n,K_{r+1})$ for large enough $n$ that is divisible by $r$. We also explore the connection to the so-called $H$-WORM colorings (colorings without rainbow or monochromatic copies of $H$) and obtain new results regarding the largest number of edges that a graph with an $H$-WORM coloring can have.