The Maximal Length of 2-Path in Random Critical Graphs

Hindawi Journal of Applied Mathematics Volume 2018, Article ID 8983218, 5 pages https://doi.org/10.1155/2018/8983218 Research Article The Maximal Length of 2-Path in Random Critical Graphs Vonjy Rasendrahasina ,1 Vlady Ravelomanana,2 and Liva Aly Raonenantsoamihaja3 1 ENS-Université d’Antananarivo, Antananarivo, Madagascar IRIF UMR CNRS 8243, Université Denis Diderot, Paris, France 3 Faculté des Sciences, Université d’Antananarivo, Antananarivo, Madagascar 2 Correspondence should be addressed to Vonjy Rasendrahasina; Received 1 December 2017; Accepted 3 April 2018; Published 14 May 2018 Academic Editor: Bruno Carpentieri 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 and 𝑀 edges uniformly randomly chosen from the set of ( 𝑛2 ) 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. 1. Preliminaries Let us recall that an undirected graph𝐺 is a couple (𝑉, 𝐸), where 𝑉 is the set of vertices and 𝐸 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 𝐺 = (𝑉, 𝐸) is sequence of vertices ⟨V0 , V1 , . . . , V𝑘 ⟩, where {V𝑖 , V𝑖+1 } ∈ 𝐸 for 𝑖 ∈ ⟦0, 𝑘 − 1⟧ and V𝑖 ≠ V𝑗 for 𝑖 ≠ 𝑗 except that its first vertex V0 might be the same as its last V𝑘 . When any two vertices of 𝐺 are connected by a path 𝐺 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 excess𝑟 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 𝐺, 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 𝐺, 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(𝑛, 𝑀 = 𝑐𝑛) is called critical if the density 𝑐 = 1/2 ± O(𝑛−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 𝜉𝑛,𝑀 of G(𝑛, 𝑀) satisfies E (𝜉𝑛,(𝑛/2)(1+𝜆𝑛−1/3 ) ) ∼ 𝑐 (𝜆) 𝑛1/3 , (1) 2 Journal of Applied Mathematics Wright [8] has shown that the EGFs (𝑊ℓ )ℓ⩾1 can be expressed in terms of 𝑇(𝑧). More precisely, Wright proved that for each ℓ ⩾ 1 there exist rational coefficients 𝑤ℓ,𝑑 , 𝑑 ∈ {0, . . . , 3ℓ + 2} such that where 𝑐 (𝜆) = 1 +∞ ∫ (1 𝛼 0 (2) 3ℓ+2 −𝑥 3𝑟 𝑊ℓ (𝑧) = ∑ − ∑√2𝜋𝑒𝑟 𝐴 (3𝑟 + 1/2, 𝜆) (1 − 𝑒 ) ) 𝑑𝑥, 𝑑=0 (1 − 𝑇 (𝑧)) 𝑟⩾0 where 𝛼 is the positive solution of 𝜆 = 𝛼−1 − 𝛼, (3) (6𝑟)! , 25𝑟 32𝑟 (3𝑟)! (2𝑟)! (4) 𝑒𝑟 is given by 𝑒𝑟 = and the function 𝐴 is defined by 𝐴 (𝑦, 𝜆) = 𝑘 ((1/2) 32/3 𝜆) −𝜆3 /6 𝑒 . ∑ 3(𝑦+1)/3 𝑘⩾0 𝑘!Γ ((𝑦 + 1 − 2𝑘) /3) (5) We remark that for Erdős-Rényi random graph G(𝑛, 𝑝 = (1 + 𝜀)/𝑛), Ding et al. [4] and Ding et al. [5] provided a complete characterisation of the structure of the giant component when 𝜀 = 𝑜(1) but 𝜀3 𝑛 → ∞. Using our notation, 𝜀 = 𝜆𝑛−1/3 but 𝜆 → ∞ as 𝑛 → ∞. 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/𝜀 = 𝜆−1 𝑛1/3 . Next, in order to reconstruct the graph, they attached trees to vertices i.i.d. Poisson(1 − 𝜀)-Galton-Watson. 2. 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. For 𝑛 ⩾ 0 and −1 ⩽ ℓ ⩽ ( 𝑛2 ), let 𝑐(𝑛, 𝑛 + ℓ) be the number of connected graphs of excess ℓ and +∞ 𝑊ℓ (𝑧) = ∑ 𝑐 (𝑛, 𝑛 + ℓ) 𝑛=0 𝑧𝑛 , 𝑛! (6) the associated EGF. We know from [7] that 𝑐(𝑛, 𝑛 − 1) = 𝑛𝑛−2 and +∞ 𝑊−1 (𝑧) = ∑ 𝑛𝑛−2 𝑛=1 𝑧𝑛 1 = 𝑇 (𝑧)2 − 𝑇 (𝑧)2 , 𝑛! 2 (7) where 𝑇(𝑧) is the EGF of rooted Cayley trees given by +∞ 𝑇 (𝑧) = 𝑧𝑒𝑇(𝑧) = ∑ 𝑛𝑛−1 𝑛=1 𝑧𝑛 . 𝑛! (8) We also have (see, e.g., [1, Equation (3.5)]) 1 1 𝑇 (𝑧) 𝑇 (𝑧)2 − − . 𝑊0 (𝑧) = log 2 1 − 𝑇 (𝑧) 2 4 𝑤ℓ,𝑑 (9) 3ℓ−𝑑 . (10) The coefficients 𝑏ℓ fl 𝑤ℓ,0 are known as Wright’s constants (see [9]). For complex graphs, denote by 𝐸𝑟 (𝑧) the EGF of these graphs of excess 𝑟. Then we have 𝐸0 (𝑧) = 1 (empty graphs) and 𝐸1 (𝑧) = 𝑊1 (𝑧). More generally, as detailed in [1, Section 8], the EGF 𝐸𝑟 (𝑧) satisfies +∞ +∞ 𝑟=0 𝑟=1 ∑ 𝐸𝑟 (𝑧) = exp ( ∑ 𝑊ℓ (𝑧)) . (11) Following [1], the EGF 𝐸𝑟 (𝑧) can also be expressed as a rational function of 𝑇(𝑧) 𝐸𝑟 (𝑧) = ∑ 𝑒𝑟,𝑑 𝑑⩾0 󸀠 𝑒𝑟,𝑑 𝑇 (𝑧)5𝑟−𝑑 (1 − 𝑇 (𝑧)) =∑ 3𝑟−𝑑 𝑑⩾0 (1 − 𝑇 (𝑧)) 3𝑟−𝑑 , (12) 󸀠 . The coefficients (𝑒𝑟 ) and (𝑏𝑟 ) are related where 𝑒𝑟 fl 𝑒𝑟,0 = 𝑒𝑟,0 by 𝑒0 = 1, 𝑟−1 𝑟𝑒𝑟 = 𝑟𝑏𝑟 + ∑𝑗𝑏𝑗 𝑒𝑟−𝑗 as 𝑟 ⩾ 1. (13) 𝑗=1 As shown in [1, 10], we remark that the dominant asymptotic behavior of [𝑧𝑛 ]𝑊ℓ (𝑧) and [𝑧𝑛 ]𝐸𝑟 (𝑧) (for any power series 𝐴(𝑧) = ∑ 𝑎𝑛 𝑧𝑛 , [𝑧𝑛 ]𝐴(𝑧) denotes the 𝑛th coefficient of 𝐴(𝑧), namely, [𝑧𝑛 ]𝐴(𝑧) = 𝑎𝑛 .) is governed by the leading coefficients 𝑏ℓ and 𝑒𝑟 . In particular, if ℓ and 𝑟 are about 𝑜(𝑛1/3 ), these EGFs satisfy 𝑊ℓ (𝑧) ≍ℓ 𝑏ℓ (1 − 𝑇 (𝑧))3ℓ , 𝑒𝑟 , 𝐸𝑟 (𝑧) ≍𝑟 (1 − 𝑇 (𝑧))3𝑟 (14) where 𝐴(𝑧) ≍ℓ 𝐵(𝑧) if and only if [𝑧𝑛 ]𝐴(𝑧) ∼ [𝑧𝑛 ]𝐵(𝑧) as 𝑛 → +∞ and ℓ = 𝑜(𝑛1/3 ). The EGF 𝑏ℓ /(1 − 𝑇(𝑧))3ℓ (resp., 𝑒𝑟 /(1 − 𝑇(𝑧))3𝑟 ) (...truncated)