[Paper Review] Singular Tur\'an numbers and WORM-colorings

Dániel Gerbner, Balázs Patkós+2 more|arXiv (Cornell University)|Sep 11, 2019

Limits and Structures in Graph Theory9 references5 citations

TL;DR

This paper determines the exact singular Turán number $T_S(n, K_3)$ for all $n \equiv 0 \pmod{4}$ and $n \equiv 1 \pmod{4}$, and establishes $T_S(n, K_{r+1}) = t'(n, r^2)$ for large $n$ divisible by $r$, resolving exact values where only asymptotics were previously known. It further connects singular Turán numbers to $H$-WORM colorings and introduces the regular Turán problem, proving a quadratic lower bound for $\mathrm{rex}(n,F)$ for non-bipartite $F$. The key contribution is closing gaps in exact singular Turán numbers and linking them to WORM colorings and regular graph extremal problems.

ABSTRACT

A subgraph $H$ of $G$ is \ extit{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.

Motivation & Objective

• To determine the exact value of the singular Turán number $T_S(n, K_3)$ for all $n \equiv 0 \pmod{4}$ and $n \equiv 1 \pmod{4}$, extending prior asymptotic results.
• To establish the exact value of $T_S(n, K_{r+1})$ for large $n$ divisible by $r$, resolving a long-standing gap in exact singular Turán numbers.
• To explore the connection between singular Turán numbers and $H$-WORM colorings, particularly in bounding the maximum number of edges in graphs with such colorings.
• To initiate the study of the regular Turán problem, defining $\mathrm{rex}(n,F)$ as the maximum number of edges in an $F$-free regular graph on $n$ vertices.

Proposed method

• Proving exact values of $T_S(n, K_3)$ via improved upper bounds using Turán's theorem and structural analysis of $K_5$-free graphs.
• Constructing extremal graphs as complete $r^2$-partite graphs with controlled part sizes to achieve $T_S(n, K_{r+1}) = t'(n, r^2)$ for large $n$ divisible by $r$.
• Using the fact that a $K_{r^2+1}$-free graph is necessary for avoiding singular $K_{r+1}$, and leveraging the Turán graph as a base construction.
• Introducing $T^*(n,r)$, a complete $r$-partite graph with parts of distinct sizes, to construct $H$-WORM colorings without monochromatic or rainbow copies of $H$.
• Adding regular $F$-free graphs within parts to preserve coloring integrity and avoid degree collisions across parts.
• Applying the Erdős–Sachs theorem to construct high-girth regular graphs, establishing a quadratic lower bound for $\mathrm{rex}(n,F)$ when $F$ has odd girth $g$.

Experimental results

Research questions

• RQ1 What is the exact value of $T_S(n, K_3)$ for $n \equiv 0 \pmod{4}$ and $n \equiv 1 \pmod{4}$, beyond the known asymptotic bounds?
• RQ2 For which $n$ and $r$ is $T_S(n, K_{r+1})$ exactly equal to $t'(n, r^2)$, the maximum number of edges in a complete $r^2$-partite graph without singular $K_{r+1}$?
• RQ3 How are singular Turán numbers related to $H$-WORM colorings, and can extremal graphs for one problem be used to bound the other?
• RQ4 What is the maximum number of edges in an $F$-free regular graph on $n$ vertices, and how does this relate to extremal graph theory?
• RQ5 Can a quadratic lower bound be established for $\mathrm{rex}(n,F)$ when $F$ is non-bipartite, and what is the dependence on the odd girth of $F$?

Key findings

• For $n = 4k$ with $k \geq 2$, $T_S(4k, K_3) = 6k^2 - 2$, and $T_S(4, K_3) = 5$, closing the gap for $n \equiv 0 \pmod{4}$.
• For $n = 4k+1$, $T_S(4k+1, K_3) = 6k^2 + 2k$, providing the exact value for this residue class.
• For $n \equiv 3 \pmod{4}$, the bounds are tightened to $6k^2 + 8k + 1 \leq T_S(4k+3, K_3) \leq 6k^2 + 8k + 3$, reducing the gap to 2.
• For large $n$ divisible by $r$, $T_S(n, K_{r+1}) = t'(n, r^2)$, where $t'(n, r^2)$ is the maximum number of edges in a complete $r^2$-partite graph with $r$ part sizes, each appearing $r$ times, and part sizes differing by at most $r$.
• The construction of $T^*(n,r)$ as a complete $r$-partite graph with distinct part sizes ensures that vertex degrees induce a valid $H$-WORM coloring, enabling new bounds on $\mathrm{wex}(n,F)$.
• For any non-bipartite $F$ with odd girth $g$, $\mathrm{rex}(n,F) \geq n^2/(g+6) - O(n)$, establishing a quadratic lower bound for the regular Turán problem.