Spanning maximal planar subgraphs of random graphs

We study the threshold for the existence of a spanning maximal planar subgraph in the random graph G_{n, p} . We show that it is very near p = 1/n^? We also discuss the threshold for the existence of a spanning maximal outerplanar subgraph. This is very near p = 1/n^½.     

Generating unlabeled connected cubic planar graphs uniformly at random

We present an expected polynomial time algorithm to generate an unlabeled connected cubic planar graph uniformly at random. We first consider rooted connected cubic planar graphs, i.e., we count connected cubic planar graphs up to isomorphisms that fix a certain directed edge. Based on decompositions along the connectivity structure, we derive recurrence formulas for the exact number of rooted cubic planar graphs. This leads to rooted 3‐connected cubic planar graphs, which have a unique embedding on the sphere. Special care has to be taken for rooted graphs that have a sense‐reversing automorphism. Therefore we introduce the concept of colored networks, which stand in bijective correspondence to rooted 3‐connected cubic planar graphs with given symmetries. Colored networks can again be decomposed along the connectivity structure. For rooted 3‐connected cubic planar graphs embedded in the plane, we switch to the dual and count rooted triangulations. Since all these numbers can be evaluated in polynomial time using dynamic programming, rooted connected cubic planar graphs can be generated uniformly at random in polynomial time by inverting the decomposition along the connectivity structure. To generate connected cubic planar graphs without a root uniformly at random, we apply rejection sampling and obtain an expected polynomial time algorithm. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Further results on random cubic planar graphs

We provide precise asymptotic estimates for the number of several classes of labeled cubic planar graphs, and we analyze properties of such random graphs under the uniform distribution. This model was first analyzed by Bodirsky and coworkers. We revisit their work and obtain new results on the enumeration of cubic planar graphs and on random cubic planar graphs. In particular, we determine the exact probability of a random cubic planar graph being connected, and we show that the distribution of the number of triangles in random cubic planar graphs is asymptotically normal with linear expectation and variance. To the best of our knowledge, this is the first time one is able to determine the asymptotic distribution for the number of copies of a fixed graph containing a cycle in classes of random planar graphs arising from planar maps.     

Largest planar matching in random bipartite graphs

For a distribution ?? over labeled bipartite (multi) graphs G = (W, M, E), |W| = |M| = n, let L(n) denote the size of the largest planar matching of G (here W and M are posets drawn on the plane as two ordered rows of nodes and edges are drawn as straight lines). We study the asymptotic (in n) behavior of L(n) for different distributions ??. Two interesting instances of this problem are Ulam's longest increasing subsequence problem and the longest common subsequence problem. We focus on the case where ?? is the uniform distribution over the k‐regular bipartite graphs on W and M. For k = o(n^1/4), we establish that $L(n) \slash \sqrt{kn}$ tends to 2 in probability when n → ∞. Convergence in mean is also studied. Furthermore, we show that if each of the n² possible edges between W and M are chosen independently with probability 0 < p < 1, then L(n)/n tends to a constant γ_p in probability and in mean when n → ∞. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 21: 162–181, 2002     

Bipolar neutrosophic planar graphs

Fuzzy graph theory is used for solving real-world problems in different fields, including theoretical computer science, engineering, physics, combinatorics and medical sciences. In this paper, we present conepts of bipolar neutrosophic multigraphs, bipolar neutrosophic planar graphs, bipolar neutrosophic dual graphs, and study some of their related properties. We also describe applications of bipolar neutrosophic graphs in road network and electrical connections.     

Group representations that resist random sampling

We show that there exists a family of groups G_n and nontrivial irreducible representations ρ_n such that, for any constant t, the average of ρ_n over t uniformly random elements has operator norm 1 with probability approaching 1 as . More quantitatively, we show that there exist families of finite groups for which random elements are required to bound the norm of a typical representation below 1. This settles a conjecture of A. Wigderson. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 605–614, 2015     

Star coloring bipartite planar graphs

A star coloring of a graph is a proper vertex‐coloring such that no path on four vertices is 2‐colored. We prove that the vertices of every bipartite planar graph can be star colored from lists of size 14, and we give an example of a bipartite planar graph that requires at least eight colors to star color. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 1–10, 2009     

Random walks on random simple graphs

This paper looks at random regular simple graphs and considers nearest neighbor random walks on such graphs. This paper considers walks where the degree d of each vertex is around (log n)^a where a is a constant which is at least 2 and where n is the number of vertices. By extending techniques of Dou, this paper shows that for most such graphs, the position of the random walk becomes close to uniformly distributed after slightly more than log n/log d steps. This paper also gets similar results for the random graph G(n, p), where p = d/(n − 1). © 1996 John Wiley & Sons, Inc.     

Grids in random graphs

Threshold probabilities for the existence in a random graph on n vertices of a graph isomorphic to a given graph of order Cn and average degree at least three are investigated. In particular it is proved that the random graph G(n, p) on n vertices with edge probability contains a square grid on En/2 vertices. © 1994 John Wiley & Sons, Inc.     

Minor‐universal planar graphs without accumulation points

We show that the countably infinite union of infinite grids, H say, is minor‐universal in the class of all graphs that can be drawn in the plane without vertex accumulation points, i.e., that H contains every such graph as a minor. Furthermore, we characterize the graphs that occur as minors of the infinite grid by a natural topological condition on their embeddings. © 2000 John Wiley & Sons, Inc. J Graph Theory 36: 1–7, 2001     

An almost linear time algorithm for finding Hamilton cycles in sparse random graphs with minimum degree at least three

We describe an algorithm for finding Hamilton cycles in random graphs. Our model is the random graph . In this model G is drawn uniformly from graphs with vertex set [n], m edges and minimum degree at least three. We focus on the case where m = cn for constant c. If c is sufficiently large then our algorithm runs in time and succeeds w.h.p. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 73–98, 2015     

Uniform random colored complexes

We present here random distributions on (D + 1)‐edge‐colored, bipartite graphs with a fixed number of vertices 2p. These graphs encode D‐dimensional orientable colored complexes. We investigate the behavior of those graphs as p→∞. The techniques involved in this study also yield a Central Limit Theorem for the genus of a uniform map of order p, as p→∞.     

List coloring triangle-free planar graphs

This paper proves the following result. Assume $G$ is a triangle-free planar graph, $X$ is an independent set of $G$. If $L$ is a list assignment of $G$ such that $\text{◂=▸}|L\left(v\right)|=4$ for each vertex $\text{◂+▸}v\in V\left(G\right)-X$ and $\text{◂=▸}|L\left(v\right)|=3$ for each vertex $v\in X$, then $G$ is $L$-colorable.     

Countable sparse random graphs

For each irrational a, 0<a<1, a particular countable graph G is defined which mirrors the asymptotic behavior of the random graph G(n, p) with edge probability p = n^?a.     

Degree distribution in random planar graphs

We prove that for each k?0, the probability that a root vertex in a random planar graph has degree k tends to a computable constant dk, so that the expected number of vertices of degree k is asymptotically dkn, and moreover that k_∑dk=1. The proof uses the tools developed by Giménez and Noy in their solution to the problem of the asymptotic enumeration of planar graphs, and is based on a detailed analysis of the generating functions involved in counting planar graphs. However, in order to keep track of the degree of the root, new technical difficulties arise. We obtain explicit, although quite involved expressions, for the coefficients in the singular expansions of the generating functions of interest, which allow us to use transfer theorems in order to get an explicit expression for the probability generating function p(w)=k_∑dkwk. From this we can compute the dk to any degree of accuracy, and derive the asymptotic estimate dk∼c⋅k−1/2qk for large values of k, where q≈0.67 is a constant defined analytically.     

Randomly coloring random graphs

We consider the problem of generating a coloring of the random graph ??_n,p uniformly at random using a natural Markov chain algorithm: the Glauber dynamics. We assume that there are βΔ colors available, where Δ is the maximum degree of the graph, and we wish to determine the least β = β(p) such that the distribution is close to uniform in O(n log n) steps of the chain. This problem has been previously studied for ??_n,p in cases where np is relatively small. Here we consider the “dense” cases, where np ε [ω ln n, n] and ω = ω(n) → ∞. Our methods are closely tailored to the random graph setting, but we obtain considerably better bounds on β(p) than can be achieved using more general techniques. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Colouring of generalized signed triangle-free planar graphs

We view an undirected graph $G$ as a symmetric digraph, where each edge $xy$ is replaced by two opposite arcs $e=(x,y)$ and $e^{?1}=(y,x)$. Assume $S$ is an inverse closed subset of permutations of positive integers. We say $G$ is $S$-$k$-colourable if for any mapping $\sigma :E\left(G\right)\to S$ with $\sigma (x,y)={\left(\sigma (y,x)\right)}^{?1}$, there is a mapping $f:V\left(G\right)\to \left[k\right]=\{1,2,\dots ,k\}$ such that ${\sigma }_e\left(f\left(x\right)\right)\ne f\left(y\right)$ for each arc $e=(x,y)$. The concept of $S$-$k$-colourable is a common generalization of several other colouring concepts. This paper is focused on finding the sets $S$ such that every triangle-free planar graph is $S$-3-colourable. Such a set $S$ is called TFP-good. Grötzsch’s theorem is equivalent to say that $S=\left\{id\right\}$ is TFP-good. We prove that for any inverse closed subset $S$ of $S_3$ which is not isomorphic to $\{id,\left(12\right)\}$, $S$ is TFP-good if and only if either $S=\left\{id\right\}$ or there exists $a\in \left[3\right]$ such that for each $\pi \in S$, $\pi \left(a\right)\ne a$. It remains an open question to determine whether or not $S=\{id,\left(12\right)\}$ is TFP-good.     

Connectivity of inhomogeneous random graphs

We find conditions for the connectivity of inhomogeneous random graphs with intermediate density. Our results generalize the classical result for G(n, p), when . We draw n independent points X_i from a general distribution on a separable metric space, and let their indices form the vertex set of a graph. An edge (i, j) is added with probability , where is a fixed kernel. We show that, under reasonably weak assumptions, the connectivity threshold of the model can be determined. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 45, 408‐420, 2014     

Saturation in random graphs

A graph H is K_s ‐saturated if it is a maximal K_s ‐free graph, i.e., H contains no clique on s vertices, but the addition of any missing edge creates one. The minimum number of edges in a K_s ‐saturated graph was determined over 50 years ago by Zykov and independently by Erd?s, Hajnal and Moon. In this paper, we study the random analog of this problem: minimizing the number of edges in a maximal K_s ‐free subgraph of the Erd?s‐Rényi random graph G (n, p ). We give asymptotically tight estimates on this minimum, and also provide exact bounds for the related notion of weak saturation in random graphs. Our results reveal some surprising behavior of these parameters. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 169–181, 2017     

Modularity in random regular graphs and lattices

Given a graph G, the modularity of a partition of the vertex set measures the extent to which edge density is higher within parts than between parts; and the modularity of G is the maximum modularity of a partition.We give an upper bound on the modularity of r-regular graphs as a function of the edge expansion (or isoperimetric number) under the restriction that each part in our partition has a sub-linear numbers of vertices. This leads to results for random r-regular graphs. In particular we show the modularity of a random cubic graph partitioned into sub-linear parts is almost surely in the interval (0.66, 0.88).The modularity of a complete rectangular section of the integer lattice in a fixed dimension was estimated in Guimer et. al. [R. Guimerà, M. Sales-Pardo and L.A. Amaral, Modularity from fluctuations in random graphs and complex networks, Phys. Rev. E 70 (2) (2004) 025101]. We extend this result to any subgraph of such a lattice, and indeed to more general graphs.