Dynamic choosability of triangle‐free graphs and sparse random graphs

Ther‐dynamic choosability of a graph G, written , is the least k such that whenever each vertex is assigned a list of at least k colors a proper coloring can be chosen from the lists so that every vertex v has at least neighbors of distinct colors. Let ch(G) denote the choice number of G. In this article, we prove when is bounded. We also show that there exists a constant C such that the random graph with almost surely satisfies . Also if G is a triangle‐free regular graph, then we have .     

Rooted induced trees in triangle‐free graphs

For a graph G, let t(G) denote the maximum number of vertices in an induced subgraph of Gthat is a tree. Further, for a vertex v∈V(G), let t(G, v) denote the maximum number of vertices in an induced subgraph of Gthat is a tree, with the extra condition that the tree must contain v. The minimum of t(G) (t(G, v), respectively) over all connected triangle‐free graphs G(and vertices v∈V(G)) on nvertices is denoted by t₃(n) (t(n)). Clearly, t(G, v)?t(G) for all v∈V(G). In this note, we solve the extremal problem of maximizing |G| for given t(G, v), given that Gis connected and triangle‐free. We show that and determine the unique extremal graphs. Thus, we get as corollary that $t_3(n)\ge t_3^{\ast}(n) = \lceil {\frac{1}{2}}(1+{\sqrt{8n-7}})\rceilFor a graph G, let t(G) denote the maximum number of vertices in an induced subgraph of Gthat is a tree. Further, for a vertex v∈V(G), let t(G, v) denote the maximum number of vertices in an induced subgraph of Gthat is a tree, with the extra condition that the tree must contain v. The minimum of t(G) (t(G, v), respectively) over all connected triangle‐free graphs G(and vertices v∈V(G)) on nvertices is denoted by t₃(n) (t(n)). Clearly, t(G, v)?t(G) for all v∈V(G). In this note, we solve the extremal problem of maximizing |G| for given t(G, v), given that Gis connected and triangle‐free. We show that and determine the unique extremal graphs. Thus, we get as corollary that $t_3(n)\ge t_3^{\ast}(n) = \lceil {\frac{1}{2}}(1+{\sqrt{8n-7}})\rceil$, improving a recent result by Fox, Loh and Sudakov. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 206–209, 2010     

Coloring triangle‐free graphs with local list sizes

We prove two distinct and natural refinements of a recent breakthrough result of Molloy (and a follow‐up work of Bernshteyn) on the (list) chromatic number of triangle‐free graphs. In both our results, we permit the amount of color made available to vertices of lower degree to be accordingly lower. One result concerns list coloring and correspondence coloring, while the other concerns fractional coloring. Our proof of the second illustrates the use of the hard‐core model to prove a Johansson‐type result, which may be of independent interest.     

Exponentially many 4‐list‐colorings of triangle‐free graphs on surfaces

Thomassen proved that every planar graph G on n vertices has at least distinct L‐colorings if L is a 5‐list‐assignment for G and at least distinct L‐colorings if L is a 3‐list‐assignment for G and G has girth at least five. Postle and Thomas proved that if G is a graph on n vertices embedded on a surface Σ of genus g, then there exist constants such that if G has an L‐coloring, then G has at least distinct L‐colorings if L is a 5‐list‐assignment for G or if L is a 3‐list‐assignment for G and G has girth at least five. More generally, they proved that there exist constants such that if G is a graph on n vertices embedded in a surface Σ of fixed genus g, H is a proper subgraph of G, and ϕ is an L‐coloring of H that extends to an L‐coloring of G, then ϕ extends to at least distinct L‐colorings of G if L is a 5‐list‐assignment or if L is a 3‐list‐assignment and G has girth at least five. We prove the same result if G is triangle‐free and L is a 4‐list‐assignment of G, where , and .     

List coloring triangle‐free hypergraphs

A triangle in a hypergraph is a collection of distinct vertices u, v, w and distinct edges e, f, g with , and . Johansson [Tech. report (1996)] proved that every triangle‐free graph with maximum degree Δ has list chromatic number . Frieze and Mubayi (Electron J Comb 15 (2008), 27) proved that every linear (meaning that every two edges share at most one vertex) triangle‐free triple system with maximum degree Δ has chromatic number . The restriction to linear triple systems was crucial to their proof. We provide a common generalization of both these results for rank 3 hypergraphs (edges have size 2 or 3). Our result removes the linear restriction from 8 , while reducing to the (best possible) result [Johansson, Tech. report (1996)] for graphs. In addition, our result provides a positive answer to a restricted version of a question of Ajtai Erd?s, Komlós, and Szemerédi (combinatorica 1 (1981), 313–317) concerning sparse 3‐uniform hypergraphs. As an application, we prove that if is the collection of 3‐uniform triangles, then the Ramsey number satisfies for some positive constants a and b. The upper bound makes progress towards the recent conjecture of Kostochka, Mubayi, and Verstraëte (J Comb Theory Ser A 120 (2013), 1491–1507) that where C₃ is the linear triangle. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 487–519, 2015     

Triangle‐free subgraphs in the triangle‐free process

Consider the triangle‐free process, which is defined as follows. Start with G(0), an empty graph on n vertices. Given G(i ‐ 1), let G(i) = G(i ‐ 1) ∪{g(i)}, where g(i) is an edge that is chosen uniformly at random from the set of edges that are not in G(i ? 1) and can be added to G(i ‐ 1) without creating a triangle. The process ends once a maximal triangle‐free graph has been created. Let H be a fixed triangle‐free graph and let X_H(i) count the number of copies of H in G(i). We give an asymptotically sharp estimate for ??(X_H(i)), for every \begin{align*}1 \ll i \le 2^{-5} n^{3/2} \sqrt{\ln n}\end{align*}, at the limit as n →∞. Moreover, we provide conditions that guarantee that a.a.s. X_H(i) = 0, and that X_H(i) is concentrated around its mean.© 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

A note on triangle‐free quasi‐symmetric designs

Triangle‐free quasi‐symmetric 2‐ (v, k, λ) designs with intersection numbers x, y; 0<x<y<kand λ>1, are investigated. It is proved that λ?2y ? x ? 3. As a consequence it is seen that for fixed λ, there are finitely many triangle‐free quasi‐symmetric designs. It is also proved that: k?y(y ? x) + x. Copyright © 2011 Wiley Periodicals, Inc. J Combin Designs 19:422‐426, 2011     

Diameter in ultra‐small scale‐free random graphs

It is well known that many random graphs with infinite variance degrees are ultra‐small. More precisely, for configuration models and preferential attachment models where the proportion of vertices of degree at least k is approximately k^{?(τ ? 1)} with τ ∈ (2,3), typical distances between pairs of vertices in a graph of size n are asymptotic to and , respectively. In this paper, we investigate the behavior of the diameter in such models. We show that the diameter is of order precisely when the minimal forward degree d_fwd of vertices is at least 2. We identify the exact constant, which equals that of the typical distances plus . Interestingly, the proof for both models follows identical steps, even though the models are quite different in nature.     

On fractional K‐factors of random graphs

Let K be a graph on r vertices and let G = (V,E) be another graph on ∣V ∣ = n vertices. Denote the set of all copies of K in G by 𝒦. A non‐negative real‐valued function f : 𝒦→ ℝ₊ is called a fractional K‐factor if ∑ _{K:v∈K∈𝒦}f(K) ≤ 1 for every v ∈ V and ∑ _K∈𝒦f(K) = n/r. For a non‐empty graph K let d(K) = e(K)/v(K) and d⁽¹⁾(K) = e(K)/(v(K) ‐ 1). We say that K is strictly K₁‐balanced if for every proper subgraph K^′ ⊊ K, d⁽¹⁾(K^′) < d⁽¹⁾(K). We say that K is imbalanced if it has a subgraph K^′ such that d(K^′) > d(K). Considering a random graph process on n vertices, we show that if K is strictly K₁‐balanced, then with probability tending to 1 as n →∞, at the first moment τ₀ when every vertex is covered by a copy of K, the graph has a fractional K‐factor. This result is the best possible. As a consequence, if K is K₁‐balanced, we derive the threshold probability function for a random graph to have a fractional K‐factor. On the other hand, we show that if K is an imbalanced graph, then for asymptotically almost every graph process there is a gap between τ₀ and the appearance of a fractional K‐factor. We also introduce and apply a criteria for perfect fractional matchings in hypergraphs in terms of expansion properties. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

The degree sequences and spectra of scale‐free random graphs

We investigate the degree sequences of scale‐free random graphs. We obtain a formula for the limiting proportion of vertices with degree d, confirming non‐rigorous arguments of Dorogovtsev, Mendes, and Samukhin ( 14 ). We also consider a generalization of the model with more randomization, proving similar results. Finally, we use our results on the degree sequence to show that for certain values of parameters localized eigenfunctions of the adjacency matrix can be found. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

On the structure of clique‐free graphs

Using a clever inductive counting argument Erd?s, Kleitman and Rothschild showed in 1976 that almost all triangle‐free graphs are bipartite, i.e., that the cardinality of the two graph classes is asymptotically equal. In this paper we investigate the structure of the “few” triangle‐free graphs which are not bipartite. As it turns out, with high probability, these graphs are bipartite up to a few vertices. More precisely, almost all of them can be made bipartite by removing just one vertex. Almost all others can be made bipartite by removing two vertices, and then three vertices and so on. We also show that similar results hold if we replace “triangle‐free” by K_??+1‐free and “bipartite” by ??‐partite. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19, 37–53, 2001     

On triangles in ‐minor free graphs

We study graphs where each edge that is incident to a vertex of small degree (of degree at most 7 and 9, respectively) belongs to many triangles (at least 4 and 5, respectively) and show that these graphs contain a complete graph (K₆ and K₇, respectively) as a minor. The second case settles a problem of Nevo. Moreover, if each edge of a graph belongs to six triangles, then the graph contains a K₈‐minor or contains K_{2, 2, 2, 2, 2} as an induced subgraph. We then show applications of these structural properties to stress freeness and coloring of graphs. In particular, motivated by Hadwiger's conjecture, we prove that every K₇‐minor free graph is 8‐colorable and every K₈‐minor free graph is 10‐colorable.     

On the max‐cut of sparse random graphs

We consider the problem of estimating the size of a maximum cut (Max‐Cut problem) in a random Erdős‐Rényi graph on n nodes and edges. It is shown in Coppersmith et al. that the size of the maximum cut in this graph normalized by the number of nodes belongs to the asymptotic region with high probability (w.h.p.) as n increases, for all sufficiently large c. The upper bound was obtained by application of the first moment method, and the lower bound was obtained by constructing algorithmically a cut which achieves the stated lower bound. In this paper, we improve both upper and lower bounds by introducing a novel bounding technique. Specifically, we establish that the size of the maximum cut normalized by the number of nodes belongs to the interval w.h.p. as n increases, for all sufficiently large c. Instead of considering the expected number of cuts achieving a particular value as is done in the application of the first moment method, we observe that every maximum size cut satisfies a certain local optimality property, and we compute the expected number of cuts with a given value satisfying this local optimality property. Estimating this expectation amounts to solving a rather involved two dimensional large deviations problem. We solve this underlying large deviation problem asymptotically as c increases and use it to obtain an improved upper bound on the Max‐Cut value. The lower bound is obtained by application of the second moment method, coupled with the same local optimality constraint, and is shown to work up to the stated lower bound value . It is worth noting that both bounds are stronger than the ones obtained by standard first and second moment methods. Finally, we also obtain an improved lower bound of on the Max‐Cut for the random cubic graph or any cubic graph with large girth, improving the previous best bound of .     

On random k‐out subgraphs of large graphs

We consider random subgraphs of a fixed graph with large minimum degree. We fix a positive integer k and let G_k be the random subgraph where each independently chooses k random neighbors, making kn edges in all. When the minimum degree then G_k is k‐connected w.h.p. for ; Hamiltonian for k sufficiently large. When , then G_k has a cycle of length for . By w.h.p. we mean that the probability of non‐occurrence can be bounded by a function (or ) where . © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 143–157, 2017     

On a 2-parameter class of scale free random graphs

A process of evolving random graphs is considered where vertices are added to the graph one by one, and edges connecting the new vertex to the old ones are drawn independently, each with probability depending linearly on the degree of the endpoint. In the paper the asymptotic degree distribution and the order of the maxdegree are determined.     

Rank‐width of random graphs

Rank‐width of a graph G, denoted by rw (G), is a width parameter of graphs introduced by Oum and Seymour [J Combin Theory Ser B 96 (2006), 514–528]. We investigate the asymptotic behavior of rank‐width of a random graph G(n, p). We show that, asymptotically almost surely, (i) if p∈(0, 1) is a constant, then rw (G(n, p)) = ?n/3??O(1), (ii) if , then rw (G(n, p)) = ?1/3??o(n), (iii) if p = c/n and c>1, then rw (G(n, p))?rn for some r = r(c), and (iv) if p?c/n and c81, then rw (G(n, p))?2. As a corollary, we deduce that the tree‐width of G(n, p) is linear in n whenever p = c/n for each c>1, answering a question of Gao [2006]. © 2011 Wiley Periodicals, Inc. J Graph Theory.     

On factors of 4‐connected claw‐free graphs*

We consider the existence of several different kinds of factors in 4‐connected claw‐free graphs. This is motivated by the following two conjectures which are in fact equivalent by a recent result of the third author. Conjecture 1 (Thomassen): Every 4‐connected line graph is hamiltonian, i.e., has a connected 2‐factor. Conjecture 2 (Matthews and Sumner): Every 4‐connected claw‐free graph is hamiltonian. We first show that Conjecture 2 is true within the class of hourglass‐free graphs, i.e., graphs that do not contain an induced subgraph isomorphic to two triangles meeting in exactly one vertex. Next we show that a weaker form of Conjecture 2 is true, in which the conclusion is replaced by the conclusion that there exists a connected spanning subgraph in which each vertex has degree two or four. Finally we show that Conjectures 1 and 2 are equivalent to seemingly weaker conjectures in which the conclusion is replaced by the conclusion that there exists a spanning subgraph consisting of a bounded number of paths © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 125–136, 2001     

Claw‐free circular‐perfect graphs

The circular chromatic number of a graph is a well‐studied refinement of the chromatic number. Circular‐perfect graphs form a superclass of perfect graphs defined by means of this more general coloring concept. This article studies claw‐free circular‐perfect graphs. First, we prove that if G is a connected claw‐free circular‐perfect graph with χ(G)>ω(G), then min{α(G), ω(G)}=2. We use this result to design a polynomial time algorithm that computes the circular chromatic number of claw‐free circular‐perfect graphs. A consequence of the strong perfect graph theorem is that minimal imperfect graphs G have min{α(G), ω(G)}=2. In contrast to this result, it is shown in Z. Pan and X. Zhu [European J Combin 29(4) (2008), 1055–1063] that minimal circular‐imperfect graphs G can have arbitrarily large independence number and arbitrarily large clique number. In this article, we prove that claw‐free minimal circular‐imperfect graphs G have min{α(G), ω(G)}≤3. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 163–172, 2010