Colorings and orientations of graphs

Bounds for the chromatic number and for some related parameters of a graph are obtained by applying algebraic techniques. In particular, the following result is proved: IfG is a directed graph with maximum outdegreed, and if the number of Eulerian subgraphs ofG with an even number of edges differs from the number of Eulerian subgraphs with an odd number of edges then for any assignment of a setS(v) ofd+1 colors for each vertexv ofG there is a legal vertex-coloring ofG assigning to each vertexv a color fromS(v).Research supported in part by a United States-Israel BSF Grant and by a Bergmann Memorial Grant.     

Colorings of plane graphs: A survey

After a brief historical account, a few simple structural theorems about plane graphs useful for coloring are stated, and two simple applications of discharging are given. Afterwards, the following types of proper colorings of plane graphs are discussed, both in their classical and choosability (list coloring) versions: simultaneous colorings of vertices, edges, and faces (in all possible combinations, including total coloring), edge-coloring, cyclic coloring (all vertices in any small face have different colors), 3-coloring, acyclic coloring (no 2-colored cycles), oriented coloring (homomorphism of directed graphs to small tournaments), a special case of circular coloring (the colors are points of a small cycle, and the colors of any two adjacent vertices must be nearly opposite on this cycle), 2-distance coloring (no 2-colored paths on three vertices), and star coloring (no 2-colored paths on four vertices). The only improper coloring discussed is injective coloring (any two vertices having a common neighbor should have distinct colors).     

Colorings of infinite graphs without one-colored rays

The R-chromatic number of a graph G is the least number of subsets of vertices forming a partition of V(G), and which induce subgraphs of G without infinite paths. For any integer n ≥ 2, we give sufficient conditions for a graph containing no subdivision of an infinite complete graph to have a R-chromatic number ≤ n. © 1996 John Wiley & Sons, Inc.     

On MAXCUT in strictly supercritical random graphs,and coloring of random graphs and random tournaments

We use a theorem by Ding, Lubetzky, and Peres describing the structure of the giant component of random graphs in the strictly supercritical regime, in order to determine the typical size of MAXCUT of in terms of ɛ. We then apply this result to prove the following conjecture by Frieze and Pegden. For every , there exists such that w.h.p. is not homomorphic to the cycle on vertices. We also consider the coloring properties of biased random tournaments. A p‐random tournament on n vertices is obtained from the transitive tournament by reversing each edge independently with probability p. We show that for the chromatic number of a p‐random tournament behaves similarly to that of a random graph with the same edge probability. To treat the case we use the aforementioned result on MAXCUT and show that in fact w.h.p. one needs to reverse edges to make it 2‐colorable.     

Colorings and interpretations: a connection between graphs and grammar forms

A mechanism for constructing families of “similar” graphs is introduced. This mechanism is essentially identical to the one considered in the theory of grammar forms and, at the same time, generalizes the notion of coloring of graphs. The mechanism applies to both directed and undirected graphs. The paper investigate hierarchies of families of graphs obtained by this mechanism, both in the directed and undirected case.     

Sobolev spaces, dimension, and random series

We investigate dimension-increasing properties of maps in Sobolev spaces; we obtain sharp results with a random process somewhat like Brownian motion.

     

Geometry of the vacant set left by random walk on random graphs,Wright's constants,and critical random graphs with prescribed degrees

We provide an explicit algorithm for sampling a uniform simple connected random graph with a given degree sequence. By products of this central result include: (1) continuum scaling limits of uniform simple connected graphs with given degree sequence and asymptotics for the number of simple connected graphs with given degree sequence under some regularity conditions, and (2) scaling limits for the metric space structure of the maximal components in the critical regime of both the configuration model and the uniform simple random graph model with prescribed degree sequence under finite third moment assumption on the degree sequence. As a substantive application we answer a question raised by ?erný and Teixeira study by obtaining the metric space scaling limit of maximal components in the vacant set left by random walks on random regular graphs.     

Symmetric spaces of graphs

Under consideration are the classes of graphs closed under modulo 2 addition, vertex permutation, and removal of isolated vertices. It is proved that there are exactly five these classes.     

Colouring random graphs

We discuss some results concerned with the behaviour of colouring algorithms on large random graphs.     

Chordal graphs and join spaces

A join space is an abstract model for partially ordered linear, spherical and projective geometries. A characterization for chordal graphs which are join spaces is given: a chordal graph is a join space if and only if it does not contain one of the two forbidden graphs as an induced subgraph.     

Hamilton cycles in random graphs and directed graphs

We prove that almost every digraph D_{2–in, 2–out} is Hamiltonian. As a corollary we obtain also that almost every graph G_4–out is Hamiltonian. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 16: 369–401, 2000     

Concentration and regularization of random graphs

This paper studies how close random graphs are typically to their expectations. We interpret this question through the concentration of the adjacency and Laplacian matrices in the spectral norm. We study inhomogeneous Erdös‐Rényi random graphs on n vertices, where edges form independently and possibly with different probabilities p_ij. Sparse random graphs whose expected degrees are fail to concentrate; the obstruction is caused by vertices with abnormally high and low degrees. We show that concentration can be restored if we regularize the degrees of such vertices, and one can do this in various ways. As an example, let us reweight or remove enough edges to make all degrees bounded above by O(d) where . Then we show that the resulting adjacency matrix concentrates with the optimal rate: . Similarly, if we make all degrees bounded below by d by adding weight d / n to all edges, then the resulting Laplacian concentrates with the optimal rate: . Our approach is based on Grothendieck‐Pietsch factorization, using which we construct a new decomposition of random graphs. We illustrate the concentration results with an application to the community detection problem in the analysis of networks. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 538–561, 2017     

Holes in random graphs

It is shown that for every >0 with the probability tending to 1 as n→∞ a random graph G(n,p) contains induced cycles of all lengths k, 3 ≤ k ≤ (1 − )n log c/c, provided c(n) = (n − 1)p(n)→∞.     

Discrepancy of random graphs and hypergraphs

Answering in a strong form a question posed by Bollobás and Scott, in this paper we determine the discrepancy between two random k‐uniform hypergraphs, up to a constant factor depending solely on k. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 47, 147–162, 2015