The Maximal Length of 2-Path in Random Critical Graphs

Abstract

Given a graph, its $2$-core is the maximal subgraph of $G$ without vertices of degree $\mathrm{1}$. A $\mathrm{2}$-path in a connected graph is a simple path in its $\mathrm{2}$-core such that all vertices in the path have degree $\mathrm{2}$, except the endpoints which have degree $⩾\mathrm{3}$. Consider the Erdős-Rényi random graph $\mathbb{G}(n,M)$ built with $n$ vertices and $M$ edges uniformly randomly chosen from the set of $\left(\begin{array}{c}n\\ 2\end{array}\right)$ edges. Let ${\xi }_{n,M}$ be the maximum $\mathrm{2}$-path length of $\mathbb{G}(n,M)$. In this paper, we determine that there exists a constant $c(\lambda )$ such that $\mathbb{E}\left({\xi }_{n,\left(n/\mathrm{2}\right)\left(\mathrm{1}+\lambda n^{-\mathrm{1}/\mathrm{3}}\right)}\right)~c(\lambda )n^{\mathrm{1}/\mathrm{3}},\hspace{0.17em}\hspace{0.17em}\mathrm{f}\mathrm{o}\mathrm{r}\hspace{0.17em}\hspace{0.17em}\mathrm{a}\mathrm{n}\mathrm{y}\hspace{0.17em}\hspace{0.17em}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}\hspace{0.17em}\hspace{0.17em}\lambda .$ This parameter is studied through the use of generating functions and complex analysis.