Singular Tur\'an numbers and WORM-colorings
D\'aniel Gerbner, Bal\'azs Patk\'os, Zsolt Tuza, M\'at\'e Vizer

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