On the size of $(K_t, K_{1,k})$-co-critical graphs

TL;DR

This paper investigates the minimum size of $(K_t, K_{1,k})$-co-critical graphs, establishing linear lower bounds on the number of edges and providing exact bounds for specific cases, advancing understanding in graph Ramsey theory.

Contribution

The paper proves linear lower bounds on the number of edges in $(K_t, K_{1,k})$-co-critical graphs and determines exact bounds for certain small cases, extending previous conjectures.

Findings

Established a linear lower bound on edges for all $t eq6$ and $k eq3$

Proved the bound is asymptotically tight for $t ext{ in }igrace 3,4,5igrace$

Derived the exact minimum edges for $(K_3, K_{1,3})$-co-critical graphs with at least 13 vertices

Abstract

Given graphs $G, H_1, H_2$, we write $G \rightarrow ({H}_1, H_2)$ if every $\{$red, blue$\}$-coloring of the edges of $G$ contains a red copy of $H_1$ or a blue copy of $H_2$. A non-complete graph $G$ is $(H_1, H_2)$-co-critical if $G \nrightarrow ({H}_1, H_2)$, but $G+e\rightarrow ({H}_1, H_2)$ for every edge $e$ in $\overline{G}$. Motivated by a conjecture of Hanson and Toft from 1987, we study the minimum number of edges over all $(K_t, K_{1,k})$-co-critical graphs on $n$ vertices. We prove that for all $t\ge3$ and $k\ge 3$, there exists a constant $\ell(t, k)$ such that, for all $n \ge (t-1)k+1$, if $G$ is a $(K_t, K_{1,k})$-co-critical graph on $n$ vertices, then $$ e(G)\ge \left(2t-4+\frac{k-1}{2}\right)n-\ell(t, k).$$ Furthermore, this linear bound is asymptotically best possible when $t\in\{3, 4,5\}$ and all $k\ge3$ and $n\ge (2t-2)k+1$. It seems non-trivial to construct extremal $(K_t, K_{1,k})$-co-critical graphs for $t\ge6$. We also obtain the sharp bound for the size of $(K_3, K_{1,3})$-co-critical graphs on $n\ge13$ vertices by showing that all such graphs have at least $3n-4$ edges.