On the Tur\'an Number of Generalized Theta Graphs

TL;DR

This paper estimates the maximum number of edges in large graphs avoiding a generalized theta graph, providing bounds based on path lengths and establishing exact results for specific cases.

Contribution

It introduces new bounds on extremal numbers for generalized theta graphs, especially when path lengths share parity and are constrained, with exact results for certain configurations.

Findings

Bounded extremal number for theta graphs with uniform parity paths

Established $O(n^{1+1/k^*})$ upper bound under specific conditions

Provided exact lower bound for $ ext{ex}(n, ext{Theta}_{3,5,5})$

Abstract

Let $\Theta_{k_1,\cdots,k_\ell}$ denote the generalized theta graph, which consists of $\ell$ internally disjoint paths with lengths $k_1,\cdots, k_{\ell}$, connecting two fixed vertices. We estimate the corresponding extremal number $\text{ex}(n,\Theta_{k_1,\cdots,k_\ell})$. When the lengths of all paths have the same parity and at most one path has length 1, $\text{ex}(n,\Theta_{k_1,\cdots,k_\ell})$ is $O(n^{1+1/k^\ast})$, where $2k^\ast$ is the length of the smallest cycle in $\Theta_{k_1,\cdots,k_\ell}$. We also establish matching lower bound in the particular case of $\text{ex}(n,\Theta_{3,5,5})$.