Abstract
Given a graph, its -core is the maximal subgraph of without vertices of degree . A -path in a connected graph is a simple path in its -core such that all vertices in the path have degree , except the endpoints which have degree . Consider the Erdős-Rényi random graph built with vertices and edges uniformly randomly chosen from the set of edges. Let be the maximum -path length of . In this paper, we determine that there exists a constant such that This parameter is studied through the use of generating functions and complex analysis.
1. 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 , where for and for except that its first vertex might be the same as its last . 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 is also called -component. A tree or acyclic component is a connected component of excess , an unicyclic component in a connected component of excess 0. If , -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 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 is called critical if the density . 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 -path length of satisfieswherewhere is the positive solution of is given by and the function is defined by
We remark that for Erdős-Rényi random graph , Ding et al. [4] and Ding et al. [5] provided a complete characterisation of the structure of the giant component when but . Using our notation, but as . They describe that the 2-core of a graph is obtained by “stretching” the edges into paths of lengths i.i.d. geometric with mean . Next, in order to reconstruct the graph, they attached trees to vertices i.i.d. -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 and , let be the number of connected graphs of excess andthe associated EGF. We know from [7] that andwhere is the EGF of rooted Cayley trees given byWe also have (see, e.g., [1, Equation (3.5)])Wright [8] has shown that the EGFs can be expressed in terms of . More precisely, Wright proved that for each there exist rational coefficients such that
The coefficients are known as Wright’s constants (see [9]). For complex graphs, denote by the EGF of these graphs of excess . Then we have (empty graphs) and . More generally, as detailed in [1, Section 8], the EGF satisfies
Following [1], the EGF can also be expressed as a rational function of where . The coefficients and are related by
As shown in [1, 10], we remark that the dominant asymptotic behavior of and (for any power series , denotes the th coefficient of , namely, .) is governed by the leading coefficients and . In particular, if and are about , these EGFs satisfywhere if and only if as and .
The EGF (resp., ) can be interpreted as EGF of connected graphs (resp., complex graphs) whose kernels are 3-regular. Such a graph has exactly (resp., ) 2-paths. A 2-path in the 2-core is enumerated by . Substituting by to obtain means attaching tree to each vertex of the 2-core. Since a 3-regular graph of excess has exactly edges, the associated graph has -paths. Note that, in the stated range, complex graphs and multigraphs of excess are both enumerated by and for large , (see [1, Equation (7.16)] and [9, Section 7]).
In our case, we need to control the length of each 2-path to a graph. We restrict our attention to complex graphs whose 2-paths are of length at most . So, instead of allowing 2-path of any length (), we use a 2-path of length at most (). The associated EGF is
3. 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 , 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 iswhere denotes the EGF of all complex components of total excess whose 2-paths are of length at most , . 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 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.
Lemma 2. Let . For any natural integers and , one has
Proof. We setFirst, using Stirling’s formula, we obtainNext, using Cauchy integral’s formula and substituting by , we obtainwhereThe contour in (26) should keep . At the critical value , we also have . This triple zero occurs in the procedure Janson et al. [1] used when investigating the value of the integral for large . Let , and let be the positive solution of (3). Following the proof of [1, Lemma 3], we will evaluate (26) on the path , where runs from to :The main contribution to the value of this integral comes from the vicinity of . The magnitude of depends on the real part of , namely, Re . Re decreases as increases and has its maximum on the circle when .
We have for uniformly in any region such that . In [1, equation (10.7)], the authors define where is the polynomialand 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 haveFor in the integrand of (26), we havewhen . Next, where the error term has been derived from those already in [1]. The proof of the lemma is completed by multiplying (25), (26), and .
Now, to complete the proof of the theorem, we use first Lemma 2 to get
Next, using Euler-Maclaurin summation, and after a change of variable ( so and ), we get
4. 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.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the French project ANR project MetACOnc, ANR-15-CE40-0014.