## Journal of Applied Mathematics

### 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 $\geqslant\mathrm{3}$. Consider the Erdős-Rényi random graph $\mathbb{G}(n,M)$ built with $n$ vertices and $M$ edges uniformly randomly chosen from the set of $(\begin{smallmatrix}n\\[5pt] 2\end{smallmatrix})$ 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 $\mathbb{E}({\xi }_{n,(n/\mathrm{2})(\mathrm{1}+\lambda {n}^{-\mathrm{1}/\mathrm{3}})})~c(\lambda ){n}^{\mathrm{1}/\mathrm{3}}, \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{a}\mathrm{n}\mathrm{y} \mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l} \lambda .$ This parameter is studied through the use of generating functions and complex analysis.

#### Article information

Source
J. Appl. Math., Volume 2018 (2018), Article ID 8983218, 5 pages.

Dates
Accepted: 3 April 2018
First available in Project Euclid: 13 June 2018

https://projecteuclid.org/euclid.jam/1528855298

Digital Object Identifier
doi:10.1155/2018/8983218

Mathematical Reviews number (MathSciNet)
MR3806007

#### Citation

Rasendrahasina, Vonjy; Ravelomanana, Vlady; Aly Raonenantsoamihaja, Liva. The Maximal Length of 2-Path in Random Critical Graphs. J. Appl. Math. 2018 (2018), Article ID 8983218, 5 pages. doi:10.1155/2018/8983218. https://projecteuclid.org/euclid.jam/1528855298

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