The Maximal Length of 2-Path in Random Critical Graphs

Copyright © 2018 Vonjy Rasendrahasina et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Given a graph, its 2-core is the maximal subgraph of G without vertices of degree 1. A 2-path in a connected graph is a simple path in its 2-core such that all vertices in the path have degree 2, except the endpoints which have degree ⩾ 3. Consider the Erdős-Rényi random graph G(n,M) built with n vertices andM edges uniformly randomly chosen from the set of ( 2 ) edges. Let ξn,M be the maximum 2-path length of G(n,M). In this paper, we determine that there exists a constant c(λ) such that E(ξn,(n/2)(1+λn−1/3)) ∼ c(λ)n1/3, for any real λ.This parameter is studied through the use of generating functions and complex analysis.


Preliminaries
Let us recall that an undirected graph is a couple (, ), where  is the set of vertices and  the set of edges, and an edge is an unordered pair of vertices.If we allow an edge between a vertex and itself (loop) or multiple edges between two vertices, we obtain a multigraph.An undirected graph without loops or multiple edges is known as a simple graph.A path in graph  = (, ) is sequence of vertices ⟨V 0 , V 1 , . . ., V  ⟩, where {V  , V +1 } ∈  for  ∈ ⟦0,  − 1⟧ and V  ̸ = V  for  ̸ =  except that its first vertex V 0 might be the same as its last V  .When any two vertices of  are connected by a path  is called connected.
A connected graph has excess if it has more edges than vertices.A connected component of excess ℓ is also called ℓ-component.A tree or acyclic component is a connected component of excess −1, an unicyclic component in a connected component of excess 0. If ℓ ⩾ 1, ℓ-components are called complex.A graph (not necessarily connected) is called complex when all its components are complex.The total excess of a graph is the number of edges plus the number of acyclic components, minus the number of vertices.In other words, the total excess of a graph is the sum of the excess of its complex components.Note that the total excess of a tree component is equal to 0 whereas its excess is equal to −1 and the total excess of a graph is nonnegative.
Given a graph , its 2-core is obtained by deleting recursively all nodes of degree 1.A 3-core or kernel of a complex graph is the graph obtained from its 2-core by repeating the following process on any vertex of degree two: for a vertex of degree two, we can remove it and splice together the two edges that it formerly touched.We observe that , its 2-core, and its kernel have the same excess.A graph is said cubic or 3-regular if all of its vertices are of degree 3. A graph is called clean if its 3-core is 3-regular (see [1]).
A random graph G(,  = ) is called critical if the density  = 1/2 ± O( −1/3 ).Such a graph contains a complex component with nonzero probability [2,3].Janson et al. [1] proved these graphs are clean (its complex components are clean) with high probability when the size of graph goes to infinity.
Theorem 1.The maximum 2-path length  , of G(, ) satisfies where where  is the positive solution of is given by and the function  is defined by We remark that for Erdős-Rényi random graph G(,  = (1 + )/), Ding et al. [4] and Ding et al. [5] provided a complete characterisation of the structure of the giant component when  = (1) but  3  → ∞.Using our notation,  =  −1/3 but  → ∞ as  → ∞.They describe that the 2core of a graph is obtained by "stretching" the edges into paths of lengths i.i.d.geometric with mean 1/ =  −1  1/3 .Next, in order to reconstruct the graph, they attached trees to vertices i.i.d.Poisson(1 − )-Galton-Watson.

Enumerative Tools
As shown in [1,3], exponential generating functions (EGFs) can lead to stringent results about the main characteristics of random graphs when they apply.Let us recall briefly the main EGFs involved in our proofs.We refer the reader to Harary and Palmer [6] for EGFs related to graphical enumeration.

Proof of Theorem 1
Consider a graph with  vertices,  edges, and a total excess .Such a graph contains exactly  −  +  tree components.They are enumerated by the following EGF: Since the total number of graphs with  vertices and  edges is ( (  2 )

𝑀
), the probability that a random (, )-graph (graphs with  vertices and  edges) is of total excess  is Similarly, the probability that a random (, )-graph is of total excess  and has no 2-path of length greater than  is where  []   () denotes the EGF of all complex components of total excess  whose 2-paths are of length at most ,  ⩾ 1.Then summing over , we get that the probability that a random (, )-graph has no 2-path of length greater than  is Following discussion in the previous section and using (15), the probability of a random P( , ⩽ ) is asymptotically equivalent to As in [11,Section 4] where Flajolet et al. described generating functions based methods to study extremal statistics on random mappings, we characterize the expectation of  , by means of truncated generating functions aforementioned.In fact, the mean value of  , is given by Then, combining (19), (20), and (21), we have To compute ( , ), we use the following lemma.
We have for ℎ( uniformly in any region such that |]| < log 2. In [1, equation (10.7)], the authors define where (, ) is the polynomial and Π() is a path in the complex plane that consists of the following three straight line segments: In particular, they proved that (, ) can be expressed as (5).
For the function (), we have For () ℎ() in the integrand of (26), we have when  = ( 1/12 ).Next, where the error term has been derived from those already in [1].The proof of the lemma is completed by multiplying ( 25 (37)

Conclusion
In this paper, we have studied the expectation of the maximal length of 2-path in random critical graph by means of enumerative and analytic combinatorics approaches when the size of the graph goes to infinity.Our analysis gives a precise description of the parameter near the critical point.