Tur\'an type problems for a fixed graph and a linear forest

TL;DR

This paper determines the exact maximum number of edges in large graphs that avoid both a fixed graph with chromatic number at least 3 and a linear forest with specific properties, advancing Turán-type extremal graph theory.

Contribution

It provides the exact Turán number for graphs avoiding a fixed high-chromatic graph and a linear forest with multiple components of size at least 3.

Findings

Exact Turán number for combined forbidden subgraphs

Characterization of extremal graphs avoiding these subgraphs

Extension of classical Turán problems to combined constraints

Abstract

Let $\mathscr{F}$ be a family of graphs. A graph $G$ is $\mathscr{F}$-free if $G$ does not contain any $F\in \mathscr{F}$ as a subgraph. The Tur\'an number, denoted by $ex(n, \mathscr{F})$, is the maximum number of edges in an $n$-vertex $\mathscr{F}$-free graph. Let $F $ be a fixed graph with $ \chi(F) \geq 3 $. A forest $H$ is called a linear forest if all components of $H$ are paths. In this paper, we determined the exact value of $ex(n, \{H, F\}) $ for a fixed graph $F$ with $\chi(F)\geq 3$ and a linear forest $H$ with at least $2$ components and each component with size at least $3$.