On splittable colorings of graphs and hypergraphs

The notion of a split coloring of a complete graph was introduced by Erd?s and Gyárfás [ 7 ] as a generalization of split graphs. In this work, we offer an alternate interpretation by comparing such a coloring to the classical Ramsey coloring problem via a two‐round game played against an adversary. We show that the techniques used and bounds obtained on the extremal (r,m)‐split coloring problem of [ 7 ] are closer in nature to the Turán theory of graphs rather than Ramsey theory. We extend the notion of these colorings to hypergraphs and provide bounds and some exact results. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 226–237, 2002     

Greedy colorings of uniform hypergraphs

We give a very short proof of an Erd?s conjecture that the number of edges in a non‐2‐colorable n‐uniform hypergraph is at least f(n)2ⁿ, where f(n) goes to infinity. Originally it was solved by József Beck in 1977, showing that f(n) at least clog n. With an ingenious recoloring idea he later proved that f(n) ≥ cn^1/3+o(1). Here we prove a weaker bound on f(n), namely f(n) ≥ cn^1/4. Instead of recoloring a random coloring, we take the ground set in random order and use a greedy algorithm to color. The same technique works for getting bounds on k‐colorability. It is also possible to combine this idea with the Lovász Local Lemma, reproving some known results for sparse hypergraphs (e.g., the n‐uniform, n‐regular hypergraphs are 2‐colorable if n ≥ 8). © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Color-bounded hypergraphs, IV: Stable colorings of hypertrees

We consider vertex colorings of hypergraphs in which lower and upper bounds are prescribed for the largest cardinality of a monochromatic subset and/or of a polychromatic subset in each edge. One of the results states that for any integers s≥2 and a≥2 there exists an integer f(s,a) with the following property. If an interval hypergraph admits some coloring such that in each edge E_i at least a prescribed number s_i≤s of colors occur and also each E_i contains a monochromatic subset with a prescribed number a_i≤a of vertices, then a coloring with these properties exists with at most f(s,a) colors. Further results deal with estimates on the minimum and maximum possible numbers of colors and the time complexity of determining those numbers or testing colorability, for various combinations of the four color bounds prescribed. Many interesting problems remain open.     

Colorings versus list colorings of uniform hypergraphs

Let $r$ be an integer with $r\ge 2$ and $G$ be a connected $r$-uniform hypergraph with $m$ edges. By refining the broken cycle theorem for hypergraphs, we show that if $k>\text{◂/▸}\frac{m-1}{\ln (1+\sqrt{2})}\approx \text{◂⋅▸}1.135\left(m-1\right)$, then the $k$-list assignment of $G$ admitting the fewest colorings is the constant list assignment. This extends the previous results of Donner, Thomassen, and the current authors for graphs.     

About equivalent interval colorings of weighted graphs

Given a graph G=(V,E) with strictly positive integer weights ω_i on the vertices iV, a k-interval coloring of G is a function I that assigns an interval I(i){1,…,k} of ω_i consecutive integers (called colors) to each vertex iV. If two adjacent vertices x and y have common colors, i.e. I(i)∩I(j)≠0/ for an edge [i,j] in G, then the edge [i,j] is said conflicting. A k-interval coloring without conflicting edges is said legal. The interval coloring problem (ICP) is to determine the smallest integer k, called interval chromatic number of G and denoted χ_int(G), such that there exists a legal k-interval coloring of G. For a fixed integer k, the k-interval graph coloring problem (k-ICP) is to determine a k-interval coloring of G with a minimum number of conflicting edges. The ICP and k-ICP generalize classical vertex coloring problems where a single color has to be assigned to each vertex (i.e., ω_i=1 for all vertices iV).Two k-interval colorings I₁ and I₂ are said equivalent if there is a permutation π of the integers 1,…,k such that ℓI₁(i) if and only if π(ℓ)I₂(i) for all vertices iV. As for classical vertex coloring, the efficiency of algorithms that solve the ICP or the k-ICP can be increased by avoiding considering equivalent k-interval colorings, assuming that they can be identified very quickly. To this purpose, we define and prove a necessary and sufficient condition for the equivalence of two k-interval colorings. We then show how a simple tabu search algorithm for the k-ICP can possibly be improved by forbidding the visit of equivalent solutions.     

Path coupling using stopping times and counting independent sets and colorings in hypergraphs

We analyse the mixing time of Markov chains using path coupling with stopping times. We apply this approach to two hypergraph problems. We show that the Glauber dynamics for independent sets in a hypergraph mixes rapidly as long as the maximum degree Δ of a vertex and the minimum size m of an edge satisfy m ≥ 2Δ + 1. We also show that the Glauber dynamics for proper q‐colorings of a hypergraph mixes rapidly if m ≥ 4 and q > Δ, and if m = 3 and q ≥ 1.65Δ. We give related results on the hardness of exact and approximate counting for both problems. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

A note on random greedy coloring of uniform hypergraphs

The smallest number of edges forming an n‐uniform hypergraph which is not r‐colorable is denoted by m(n,r). Erd?s and Lovász conjectured that . The best known lower bound was obtained by Radhakrishnan and Srinivasan in 2000. We present a simple proof of their result. The proof is based on the analysis of a random greedy coloring algorithm investigated by Pluhár in 2009. The proof method extends to the case of r‐coloring, and we show that for any fixed r we have improving the bound of Kostochka from 2004. We also derive analogous bounds on minimum edge degree of an n‐uniform hypergraph that is not r‐colorable. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 407–413, 2015     

List colorings of multipartite hypergraphs

Let χ_l(G) denote the list chromatic number of the r‐uniform hypergraph G. Extending a result of Alon for graphs, Saxton and the second author used the method of containers to prove that, if G is simple and d‐regular, then . To see how close this inequality is to best possible, we examine χ_l(G) when G is a random r‐partite hypergraph with n vertices in each class. The value when r = 2 was determined by Alon and Krivelevich; here we show that almost surely, where d is the expected average degree of G and . The function g(r,α) is defined in terms of “preference orders” and can be determined fairly explicitly. This is enough to show that the container method gives an optimal lower bound on χ_l(G) for r = 2 and r = 3, but, perhaps surprisingly, apparently not for r ≥ 4.     

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     

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     

Oriented list colorings of graphs

A 2‐assignment on a graph G = (V,E) is a collection of pairs L(v) of allowed colors specified for all vertices v ∈V. The graph G (with at least one edge) is said to have oriented choice number 2 if it admits an orientation which satisfies the following property: For every 2‐assignment there exists a choice c(v)∈L(v) for all v ∈V such that (i) if c(v) = c(w), then vw ∉ E, and (ii) for every ordered pair (a,b) of colors, if some edge oriented from color a to color b occurs, then no edge is oriented from color b to color a. In this paper we characterize the following subclasses of graphs of oriented choice number 2: matchings; connected graphs; graphs containing at least one cycle. In particular, the first result (which implies that the matching with 11 edges has oriented choice number 2) proves a conjecture of Sali and Simonyi. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 217–229, 2001     

Packing perfect matchings in random hypergraphs

We introduce a new procedure for generating the binomial random graph/hypergraph models, referred to as online sprinkling. As an illustrative application of this method, we show that for any fixed integer , the binomial ‐uniform random hypergraph contains edge‐disjoint perfect matchings, provided , where is an integer depending only on . Our result for is asymptotically optimal and for is optimal up to the factor. This significantly improves a result of Frieze and Krivelevich.     

On hamiltonian colorings for some graphs

For a connected graph G and any two vertices u and v in G, let D(u,v) denote the length of a longest u-v path in G. A hamiltonian coloring of a connected graph G of order n is an assignment c of colors (positive integers) to the vertices of G such that |c(u)−c(v)|+D(u,v)≥n−1 for every two distinct vertices u and v in G. The value of a hamiltonian coloring c is the maximum color assigned to a vertex of G. The hamiltonian chromatic number of G is taken over all hamiltonian colorings c of G. In this paper we discuss the hamiltonian chromatic number of graphs G with . As examples, we determine the hamiltonian chromatic number for a class of caterpillars, and double stars.     

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.     

The analytic connectivity in uniform hypergraphs: Properties and computation

The analytic connectivity (AC), defined via solving a series of constrained polynomial optimization problems, serves as a measure of connectivity in hypergraphs. How to compute such a quantity efficiently is important in practice and of theoretical challenge as well due to the non-convex and combinatorial features in its definition. In this article, we first perform a careful analysis of several widely used structured hypergraphs in terms of their properties and heuristic upper bounds of ACs. We then present an affine-scaling method to compute some upper bounds of ACs for uniform hypergraphs. To testify the tightness of the obtained upper bounds, two possible approaches via the Pólya theorem and semidefinite programming respectively are also proposed to verify the lower bounds generated by the obtained upper bounds minus a small gap. Numerical experiments on synthetic datasets are reported to demonstrate the efficiency of our proposed method. Further, we apply our method in hypergraphs constructed from social networks and text analysis to detect the network connectivity and rank the keywords, respectively.