Tur\'{a}n number for odd-ballooning of trees

TL;DR

This paper determines the maximum number of edges in large graphs avoiding the odd-ballooning of a tree, extending known results and providing counterexamples to a conjecture on monochromatic subgraphs in edge-colored complete graphs.

Contribution

It exactly computes Turán numbers for the odd-ballooning of trees when the graph is large and the structure is 'good', generalizing previous specific cases and challenging a longstanding conjecture.

Findings

Exact Turán number for large n and good T_o

Generalizes results for star trees

Provides counterexamples to Keevash and Sudakov's conjecture

Abstract

The Tur\'an number $ex(n,H)$ is the maximum number of edges in an $H$-free graph on $n$ vertices. Let $T$ be any tree. The odd-ballooning of $T$, denoted by $T_o$, is a graph obtained by replacing each edge of $T$ with an odd cycle containing the edge, and all new vertices of the odd cycles are distinct. In this paper, we determine the exact value of $ex(n,T_o)$ for sufficiently large $n$ and $T_o$ being good, which generalizes all the known results on $ex(n,T_o)$ for $T$ being a star, due to Erd\H{o}s et al. (1995), Hou et al. (2018) and Yuan (2018), and provides some counterexamples with chromatic number 3 to a conjecture of Keevash and Sudakov (2004), on the maximum number of edges not in any monochromatic copy of $H$ in a $2$-edge-coloring of a complete graph of order $n$.