Tur\'an numbers of theta graphs

TL;DR

This paper determines upper bounds on the maximum number of edges in graphs avoiding certain theta subgraphs, specifically for fixed odd path lengths and large numbers of disjoint paths, establishing tight bounds up to a constant.

Contribution

It provides new extremal bounds for theta graphs with fixed odd path length and many disjoint paths, extending Turán-type results in extremal graph theory.

Findings

Maximum edges in graphs avoiding $ heta_{ ext{odd}, t}$ are bounded by $c_{ ext{odd}} t^{1-1/ ext{odd}} n^{1+1/ ext{odd}}$

Bounds are tight up to a constant factor for large $t$ and fixed odd $ ext{ell}$

The results extend classical Turán problems to theta graphs with multiple disjoint paths.

Abstract

The theta graph $\Theta_{\ell,t}$ consists of two vertices joined by $t$ vertex-disjoint paths of length $\ell$ each. For fixed odd $\ell$ and large $t$, we show that the largest graph not containing $\Theta_{\ell,t}$ has at most $c_{\ell} t^{1-1/\ell}n^{1+1/\ell}$ edges and that this is tight apart from the value of $c_{\ell}$.