On the Size of $(K_t, P_k)$-Co-Critical Graphs

Abstract

Given graphs $G, H_1, \ldots , H_k$, we write $G \rightarrow ({H}_1, \ldots , H_k)$ if every k-coloring $\tau :E(G)\rightarrow [k]$ contains a monochromatic copy of $H_i$ in color i for some color $i\in [k]$, where $[k]=\{1, \ldots , k\}$. A non-complete graph G is $(H_1, \ldots , H_k)$-co-critical if $G \nrightarrow ({H}_1, \ldots , H_k)$, but $G+e\rightarrow ({H}_1,\ldots , H_k)$ for every edge e in $\overline{G}$. Motivated by a conjecture of Hanson and Toft from 1987, there are now a variety of papers studying the minimum number of possible edges over all $(H_1, \ldots , H_k)$-co-critical graphs. It seems non-trivial to construct extremal $(H_1, \ldots , H_k)$-co-critical graphs in general. In this paper, we focus on the study of the minimum number of edges over all $(K_t, P_k)$-co-critical graphs, where $P_k$ denotes the path on $k\ge 3$ vertices. Using the minimum path cover of a graph, we prove that for $t\in \{3,4,5\}$ and all $k\ge 4$, there exists a constant C(t, k) such that, for all $n\ge (2t-3)(k-1)+{\lceil k/2 \rceil }({\lceil k/2 \rceil }-1)+1$, there exists a $(K_t, P_k)$-co-critical graph G on n vertices satisfying

$$\begin{aligned} e(G)\le \left( \frac{4t-9}{2} + \frac{1}{2} \left\lceil \frac{k}{2} \right\rceil \right) n + C(t,k). \end{aligned}$$

We then establish the sharp bound for the size of $(K_3, P_4)$-co-critical graphs on $n\ge 14$ vertices by showing that all such graphs have at least $\lfloor (5n-3)/2\rfloor$ edges.