PARTIAL RESULT ON HADWIGER'S CONJECTURE

Conditions for [formula omitted]-graphic sequences to be potentially [formula omitted]-graphic

Degree-based graph construction is a fundamental problem in network science. A graph is simple if there are no self-loops and no multiple links between any pair of nodes in the graph. A degree sequence d = ( d 1 , d 2 , ⋯ , d N ) is graphical if d can be represented as the degree sequence of at least one simple graph, where the graph is called a realization of the sequence d. In this work, we introduce a novel method (LSFGR) for generating simple graphs from graphical degree sequences, focusing additionally on connectedness and on assortativity. LSFGR guarantees connected graphs for all potentially connected degree sequences. In the case where a degree sequence has no simple realization, LSFGR produces graphs with at most one node with self-loops. In addition, the graphs generated from LSFGR characterize real-world networks with medium assortativity.

The distance degree sequence and distance neighborhood degree sequence are two sequences in fuzzy graphs. In graph theory, the distance degree sequence of a vertex involves the concept of the number of vertices at various distances from the given vertex of a graph. The concept of distance degree sequence of a vertex in fuzzy graph is developed by taking into account the membership values of the edges which are incident on the vertices at various distances from the given vertex of fuzzy graphs, and the distance neighborhood degree sequence considers the membership values of the vertices at various distances. The drastic sum of fuzzy graphs is an operation in fuzzy graph theory, which is the extension of the standard operations on graphs using the drastic sum of fuzzy sets. The drastic sum provides a way to combine two fuzzy graphs into a new fuzzy graph by using a specific rule for combining the membership values of edges and vertices between two fuzzy graphs. In this paper, the distance degree sequence and distance neighborhood degree sequence of the vertices in the drastic sum of two fuzzy graphs on star graphs are obtained in terms of the parameters of the given graphs by considering all possible cases of drastic sum in detail.

Large graphs are sometimes studied through their degree sequences (power law or regular graphs). We study graphs that are uniformly chosen with a given degree sequence. Under mild conditions, it is shown that sequences of such graphs have graph limits in the sense of Lov\'{a}sz and Szegedy with identifiable limits. This allows simple determination of other features such as the number of triangles. The argument proceeds by studying a natural exponential model having the degree sequence as a sufficient statistic. The maximum likelihood estimate (MLE) of the parameters is shown to be unique and consistent with high probability. Thus $n$ parameters can be consistently estimated based on a sample of size one. A fast, provably convergent, algorithm for the MLE is derived. These ingredients combine to prove the graph limit theorem. Along the way, a continuous version of the Erd\H{o}s--Gallai characterization of degree sequences is derived.

Difference graphs

Let A = (a1, a2, ..., an) be a degree sequence of a simple bipartite graph. We present an algorithm that takes A as input, and outputs a simple bipartite realization of A, without stalling. The running time of the algorithm is ⊝(n1n2), where ni is the number of vertices in the part i of the bipartite graph. Then we couple the generation algorithm with a rejection sampling scheme to generate a simple realization of A uniformly at random. The best algorithm we know is the implicit one due to Bayati, Kim and Saberi (2010) that has a running time of O(mamax), where m = 1 2 ∑ i = 1 n a i $m = {1 \over 2}\sum\nolimits_{i = 1}^n {{a_i}} and amax is the maximum of the degrees, but does not sample uniformly. Similarly, the algorithm presented by Chen et al. (2005) does not sample uniformly, but nearly uniformly. The realization of A output by our algorithm may be a start point for the edge-swapping Markov Chains pioneered by Brualdi (1980) and Kannan et al.(1999).

A simple graph G is a 2-tree if G = K3, or G has a vertex v of degree 2, whose neighbors are adjacent, and G — v is a 2-tree. Clearly, if G is a 2-tree on n vertices, then E(G) = 2n — 3. A non-increasing sequence π = (d1,...,dn) of nonnegative integers is a graphic sequence if it is realizable by a simple graph G on n vertices. [Acta Math. Sin. Engl. Ser., 25, 795–802 (2009)] proved that if k ≥ 2, n ≥ $$\frac{9}{2}{k^2} + \frac{{19}}{2}k$$ and π = (d1,...,dn) is a graphic sequence with $$\sum\nolimits_{i = 1}^n {{d_i}} > (k - 2)n$$, then p has a realization containing every 1-tree (the usual tree) on k vertices. Moreover, the lower bound (k — 2)n is the best possible. This is a variation of a conjecture due to Erdos and Sos. In this paper, we investigate an analogue problem for 2-trees and prove that if k ≥ 3 is an integer with k = i (mod3), $$k \equiv i(\bmod 3)n \ge 20{[\frac{k}{3}]^2} + 31[\frac{k}{3}] + 12$$ π = (d1,...,dn is a graphic sequence with $$\sum\nolimits_{i = 1}^n {{d_i}} > \max \{ (k - 1)(n - 1),2[\frac{k}{3}]n - 2n - {[\frac{k}{3}]^2} + [\frac{{2k}}{3}] + 1 - 1{( - 1)^i}\} $$, then π has a realization containing every 2-tree on k vertices. Moreover, the lower bound max $$\{ k - 1)(n - 1),2[\frac{k}{3}]n - 2n - {[\frac{{2k}}{3}]^2} + [\frac{{2k}}{3}] + 1 - 1{( - 1)^i}\} $$ is the best possible. This result implies a conjecture due to [Discrete Math. Theor. Comput. Sci., 17(3), 315–326 (2016)].

Bounding the number of cycles in a graph in terms of its degree sequence

Given the degree sequence d of a graph, the realization graph of d is the graph having as its vertices the labeled realizations of d, with two vertices adjacent if one realization may be obtained from the other via an edge-switching operation. We describe a connection between Cartesian products in realization graphs and the canonical decomposition of degree sequences described by R.I. Tyshkevich and others. As applications, we characterize the degree sequences whose realization graphs are triangle-free graphs or hypercubes.

We investigate the effect of a fixed forbidden clique minor upon the strong chromatic index, both in multigraphs and in simple graphs. We conjecture for each that any ‐minor‐free multigraph of maximum degree has strong chromatic index at most . We present a construction certifying that if true the conjecture is asymptotically sharp as . In support of the conjecture, we show it in the case and prove the statement for strong clique number in place of strong chromatic index. By contrast, we make a basic observation that for ‐minor‐free simple graphs, the problem of strong edge‐colouring is “between” Hadwiger's Conjecture and its fractional relaxation. For , we also show that ‐minor‐free multigraphs of edge‐diameter at most 2 have strong clique number at most .

Two sufficient conditions for a graphic sequence to have a realization with prescribed clique size

Uniform sampling from graphical realizations of a given degree sequence is a fundamental component in simulation-based measurements of network observables, with applications ranging from epidemics, through social networks to Internet modeling. Existing graph sampling methods are either link-swap based (Markov-Chain Monte Carlo algorithms) or stub-matching based (the Configuration Model). Both types are ill-controlled, with typically unknown mixing times for link-swap methods and uncontrolled rejections for the Configuration Model. Here we propose an efficient, polynomial time algorithm that generates statistically independent graph samples with a given, arbitrary, degree sequence. The algorithm provides a weight associated with each sample, allowing the observable to be measured either uniformly over the graph ensemble, or, alternatively, with a desired distribution. Unlike other algorithms, this method always produces a sample, without back-tracking or rejections. Using a central limit theorem-based reasoning, we argue, that for large , and for degree sequences admitting many realizations, the sample weights are expected to have a lognormal distribution. As examples, we apply our algorithm to generate networks with degree sequences drawn from power-law distributions and from binomial distributions.

Vertex-weighted realizations of graphs

In this paper, we give structural and degree sequence characterizations for a new class of digraphs called directed 3-cycle anchored. A digraph in this class has the property that, for every realization of its degree sequence, there is a directed 3-cycle through each vertex of a labeled vertex set. We end by illustrating their use in the uniform sampling of simple directed graph realizations from a fixed degree sequence.

We prove that if G runs over the set of graphs with a fixed degree sequence d, then the values χ(G) of the function chromatic number completely cover a line segment [a,b] of positive integers. Thus for an arbitrary graphical sequence d, two invariants minχ(d):=a and maxχ(d):=b naturally arise. For a regular graphical sequence d=rn:=(r,r,…,r) where r is the degree and n is the number of vertices, the exact values of a and b are found in all situations, except the case where n and r are both even and n<2r.