Some exact results of the generalized Tur\'an numbers for paths

TL;DR

This paper establishes exact results for generalized Turán numbers involving paths and graphs with specific chromatic properties, confirming several conjectures for bipartite graphs and small path lengths.

Contribution

It proves that certain pairs of graphs are strictly Turán-good under specified conditions, confirming conjectures for bipartite graphs with matching number constraints and small path lengths.

Findings

$(H, F)$ is strictly Turán-good for bipartite $H$ with matching number $ u(H)=loor{|V(H)|/2}$ and $oxed{ ext{chromatic number } ext{ } oxed{3}}$.

$(P_ ext{ extcolor{red}{ extbf{l}}}, F)$ is strictly Turán-good for $2 extcolor{red}{ extbf{ ext{ to }}} 6$ and $oxed{ ext{chromatic number } extcolor{red}{ extbf{ extgreater} 3}}$.

Confirms conjectures about Turán-goodness for specific graph pairs and path lengths.

Abstract

For graphs $H$ and $F$ with chromatic number $\chi(F)=k$, we call $H$ strictly $F$-Tur\'an-good (or $(H, F)$ strictly Tur\'an-good) if the Tur\'an graph $T_{k-1}(n)$ is the unique $F$-free graph on $n$ vertices containing the largest number of copies of $H$ when $n$ is large enough. Let $F$ be a graph with chromatic number $\chi(F)\geq 3$ and a color-critical edge and let $P_\ell$ be a path with $\ell$ vertices. Gerbner and Palmer (2020, arXiv:2006.03756) showed that $(P_3, F)$ is strictly Tur\'an good if $\chi(H)\ge 4$ and they conjectured that (a) this result is true when $\chi(F)=3$, and, moreover, (b) $(P_\ell, K_k)$ is Tur\'an-good for every pair of integers $\ell$ and $k$. In the present paper, we show that $(H, F)$ is strictly Tur\'an-good when $H$ is a bipartite graph with matching number $\nu(H)=\lfloor \frac{|V(H)|}{2}\rfloor$ and $\chi(F)= 3$, as a corollary, this result confirms the conjecture (a); we also prove that $(P_\ell, F)$ is strictly Tur\'an-good for $2\le\ell\leq 6$ and $\chi(F)\ge 4$, this also confirms the conjecture (b) for $2\le\ell\leq 6$ and $k\ge 4$.