Acyclic edge chromatic number of outerplanar graphs

A proper edge coloring of a graph G is called acyclic if there is no 2‐colored cycle in G. The acyclic edge chromatic number of G, denoted by χ(G), is the least number of colors in an acyclic edge coloring of G. In this paper, we determine completely the acyclic edge chromatic number of outerplanar graphs. The proof is constructive and supplies a polynomial time algorithm to acyclically color the edges of any outerplanar graph G using χ(G) colors. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 22–36, 2010     

Flows and Bisections in Cubic Graphs

A k‐weak bisection of a cubic graph G is a partition of the vertex‐set of G into two parts V₁ and V₂ of equal size, such that each connected component of the subgraph of G induced by () is a tree of at most vertices. This notion can be viewed as a relaxed version of nowhere‐zero flows, as it directly follows from old results of Jaeger that every cubic graph G with a circular nowhere‐zero r‐flow has a ‐weak bisection. In this article, we study problems related to the existence of k‐weak bisections. We believe that every cubic graph that has a perfect matching, other than the Petersen graph, admits a 4‐weak bisection and we present a family of cubic graphs with no perfect matching that do not admit such a bisection. The main result of this article is that every cubic graph admits a 5‐weak bisection. When restricted to bridgeless graphs, that result would be a consequence of the assertion of the 5‐flow Conjecture and as such it can be considered a (very small) step toward proving that assertion. However, the harder part of our proof focuses on graphs that do contain bridges.     

Nowhere‐zero flows in tensor product of graphs

In this paper, we characterize graphs whose tensor product admit nowhere‐zero 3‐flow. The main result is: For two graphs G₁ and G₂ with δ ≥ 2 and G₂ not belonging to a well‐characterized class of graphs, the tensor product of G₁ and G₂ admits a nowhere‐zero 3‐flow. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 284–292, 2007     

The determining number of a Cartesian product

A set S of vertices is a determining set for a graph G if every automorphism of G is uniquely determined by its action on S. The determining number of G, denoted Det(G), is the size of a smallest determining set. This paper begins by proving that if G=G□?□G is the prime factor decomposition of a connected graph then Det(G)=max{Det(G)}. It then provides upper and lower bounds for the determining number of a Cartesian power of a prime connected graph. Further, this paper shows that Det(Q_n)=?log₂n?+1 which matches the lower bound, and that Det(K)=?log₃(2n+1)?+1 which for all n is within one of the upper bound. The paper concludes by proving that if H is prime and connected, Det(Hⁿ)=Θ(logn). © 2009 Wiley Periodicals, Inc. J Graph Theory     

The circular chromatic number of the Mycielskian of G

In a search for triangle-free graphs with arbitrarily large chromatic numbers, Mycielski developed a graph transformation that transforms a graph G into a new graph μ(G), we now call the Mycielskian of G, which has the same clique number as G and whose chromatic number equals χ(G) + 1. Chang, Huang, and Zhu [G. J. Chang, L. Huang, & X. Zhu, Discrete Math, to appear] have investigated circular chromatic numbers of Mycielskians for several classes of graphs. In this article, we study circular chromatic numbers of Mycielskians for another class of graphs G. The main result is that χ_c(μ(G)) = χ(μ(G)), which settles a problem raised in [G. J. Chang, L. Huang, & X. Zhu, Discrete Math, to appear, and X. Zhu, to appear]. As χ_c(G) = and χ(G) = , consequently, there exist graphs G such that χ_c(G) is as close to χ(G) − 1 as you want, but χ_c(μ(G)) = χ(μ(G)). © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 63–71, 1999     

Graph minors and linkages

Bollobás and Thomason showed that every 22k‐connected graph is k‐linked. Their result used a dense graph minor. In this paper, we investigate the ties between small graph minors and linkages. In particular, we show that a 6‐connected graph with a K minor is 3‐linked. Further, we show that a 7‐connected graph with a K minor is (2,5)‐linked. Finally, we show that a graph of order n and size at least 7n?29 contains a K minor. © 2005 Wiley Periodicals, Inc. J Graph Theory 49: 75–91, 2005     

An extremal problem for subdivisions of K

It is proved that every graph G with ‖G‖ ≥ 2|G| − 5, |G| ≥ 6, and girth at least 5, except the Petersen graph, contains a subdivision of K, the complete graph on five vertices minus one edge. © 1999 John Wiley & Sons, Inc, J. Graph Theory 30: 261–276, 1999     

Contractible elements in k‐connected graphs not containing some specified graphs

In 15 , Thomassen proved that any triangle‐free k‐connected graph has a contractible edge. Starting with this result, there are several results concerning the existence of contractible elements in k‐connected graphs which do not contain specified subgraphs. These results extend Thomassen's result, cf., 2 , 3 , 9 - 13 . In particular, Kawarabayashi 12 proved that any k‐connected graph without K subgraphs contains either a contractible edge or a contractible triangle. In this article, we further extend these results, and prove the following result. Let k be an integer with k ≥ 6. If G is a k‐connected graph such that G does not contain as a subgraph and G does not contain as an induced subgraph, then G has either a contractible edge which is not contained in any triangle or a contractible triangle. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:97–109, 2008     

Word maps and spectra of random graph lifts

We study here the spectra of random lifts of graphs. Let G be a finite connected graph, and let the infinite tree T be its universal cover space. If λ₁ and ρ are the spectral radii of G and T respectively, then, as shown by Friedman (Graphs Duke Math J 118 (2003), 19–35), in almost every n‐lift H of G, all “new” eigenvalues of H are ≤ O(λ ρ^1/2). Here we improve this bound to O(λ ρ^2/3). It is conjectured in (Friedman, Graphs Duke Math J 118 (2003) 19–35) that the statement holds with the bound ρ + o(1) which, if true, is tight by (Greenberg, PhD thesis, 1995). For G a bouquet with d/2 loops, our arguments yield a simple proof that almost every d‐regular graph has second eigenvalue O(d^2/3). For the bouquet, Friedman (2008). has famously proved the (nearly?) optimal bound of . Central to our work is a new analysis of formal words. Let w be a formal word in letters g,…,g. The word map associated with w maps the permutations σ₁,…,σ_k ∈ S_n to the permutation obtained by replacing for each i, every occurrence of g_i in w by σ_i. We investigate the random variable X that counts the fixed points in this permutation when the σ_i are selected uniformly at random. The analysis of the expectation ??(X) suggests a categorization of formal words which considerably extends the dichotomy of primitive vs. imprimitive words. A major ingredient of a our work is a second categorization of formal words with the same property. We establish some results and make a few conjectures about the relation between the two categorizations. These conjectures suggest a possible approach to (a slightly weaker version of) Friedman's conjecture. As an aside, we obtain a new conceptual and relatively simple proof of a theorem of A. Nica (Nica, Random Struct Algorithms 5 (1994), 703–730), which determines, for every fixed w, the limit distribution (as n →∞) of X. A surprising aspect of this theorem is that the answer depends only on the largest integer d so that w = u^d for some word u. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

Intersections of graphs

The study of graph homomorphisms has a long and distinguished history, with applications in many areas of graph theory. There has been recent interest in counting homomorphisms, and in particular on the question of finding upper bounds for the number of homomorphisms from a graph G into a fixed image graph H. We introduce our techniques by proving that the lex graph has the largest number of homomorphisms into K₂ with one looped vertex (or equivalently, the largest number of independent sets) among graphs with fixed number of vertices and edges. Our main result is the solution to the extremal problem for the number of homomorphisms into P, the completely looped path of length 2 (known as the Widom–Rowlinson model in statistical physics). We show that there are extremal graphs that are threshold, give explicitly a list of five threshold graphs from which any threshold extremal graph must come, and show that each of these “potentially extremal” threshold graphs is in fact extremal for some number of edges. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory 67: 261–284, 2011     

Circuit Covers of Signed Graphs

We introduce the concept of a signed circuit cover of a signed graph. A signed circuit cover is a natural analog of a circuit cover of a graph and is equivalent to a covering of the corresponding signed graphic matroid with circuits. As in the case of graphs, a signed graph has a signed circuit cover only when it admits a nowhere‐zero integer flow. In the present article, we establish the existence of a universal coefficient such that every signed graph G that admits a nowhere‐zero integer flow has a signed circuit cover of total length at most . We show that if G is bridgeless, then , and in the general case .     

First order zero–one laws for random graphs on the circle

We look at a competitor of the Erdős–Rényi models of random graphs, one proposed in E. Gilbert [J. Soc. Indust. Appl. Math. 9:4, 533–543 (1961)]: given δ>0 and a metric space X of diameter >δ, scatter n vertices at random on X and connect those of distance <δ apart: we get a random graph G. Letting X be a circle, we look at zero‐one laws for (in First Order Logic) various δ. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 14, 239–266, 1999     

The rotational dimension of a graph

Given a connected graph G = (N, E) with node weights s∈? and nonnegative edge lengths, we study the following embedding problem related to an eigenvalue optimization problem over the second smallest eigenvalue of the (scaled) Laplacian of G: Find v_i∈?^|N|, i∈N so that distances between adjacent nodes do not exceed prescribed edge lengths, the weighted barycenter of all points is at the origin, and is maximized. In the case of a two‐dimensional optimal solution this corresponds to the equilibrium position of a quickly rotating net consisting of weighted mass points that are linked by massless cables of given lengths. We define the rotational dimension of G to be the minimal dimension k so that for all choices of lengths and weights an optimal solution can be found in ?^k and show that this is a minor monotone graph parameter. We give forbidden minor characterizations up to rotational dimension 2 and prove that the rotational dimension is always bounded above by the tree‐width of G plus one. © 2010 Wiley Periodicals, Inc. J Graph Theory 66:283‐302, 2011     

Properly colored subgraphs and rainbow subgraphs in edge‐colorings with local constraints

We consider a canonical Ramsey type problem. An edge‐coloring of a graph is called m‐good if each color appears at most m times at each vertex. Fixing a graph G and a positive integer m, let f(m, G) denote the smallest n such that every m‐good edge‐coloring of K_n yields a properly edge‐colored copy of G, and let g(m, G) denote the smallest n such that every m‐good edge‐coloring of K_n yields a rainbow copy of G. We give bounds on f(m, G) and g(m, G). For complete graphs G = K_t, we have c₁mt²/ln t ≤ f(m, K_t) ≤ c₂mt², and cmt³/ln t ≤ g(m, K_t) ≤ cmt³/ln t, where c₁, c₂, c, c are absolute constants. We also give bounds on f(m, G) and g(m, G) for general graphs G in terms of degrees in G. In particular, we show that for fixed m and d, and all sufficiently large n compared to m and d, f(m, G) = n for all graphs G with n vertices and maximum degree at most d. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 2003     

On the density of 2‐colorable 3‐graphs in which any four points span at most two edges

Let ex₂(n, K) be the maximum number of edges in a 2‐colorable K‐free 3‐graph (where K={123, 124, 134} ). The 2‐chromatic Turán density of K is $\pi_{2}({K}_{4}^-) =lim_{{n}\to \infty} {ex}_{2}({n}, {K}_{4}^-)/\left(_{3}^{n}\right)Let ex₂(n, K) be the maximum number of edges in a 2‐colorable K‐free 3‐graph (where K={123, 124, 134} ). The 2‐chromatic Turán density of K is $\pi_{2}({K}_{4}^-) =lim_{{n}\to \infty} {ex}_{2}({n}, {K}_{4}^-)/\left(_{3}^{n}\right)$. We improve the previously best known lower and upper bounds of 0.25682 and 3/10?ε, respectively, by showing that This implies the following new upper bound for the Turán density of K In order to establish these results we use a combination of the properties of computer‐generated extremal 3‐graphs for small n and an argument based on “super‐saturation”. Our computer results determine the exact values of ex(n, K) for n≤19 and ex₂(n, K) for n≤17, as well as the sets of extremal 3‐graphs for those n. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 105–114, 2010     

Circular chromatic index of graphs of maximum degree 3

This paper proves that if G is a graph (parallel edges allowed) of maximum degree 3, then χ′_c(G) ≤ 11/3 provided that G does not contain H₁ or H₂ as a subgraph, where H₁ and H₂ are obtained by subdividing one edge of K (the graph with three parallel edges between two vertices) and K₄, respectively. As χ′_c(H₁) = χ′_c(H₂) = 4, our result implies that there is no graph G with 11/3 < χ′_c(G) < 4. It also implies that if G is a 2‐edge connected cubic graph, then χ′_c(G) ≤ 11/3. © 2005 Wiley Periodicals, Inc. J Graph Theory 49: 325–335, 2005     

On Cubic Bridgeless Graphs Whose Edge‐Set Cannot be Covered by Four Perfect Matchings

The problem of establishing the number of perfect matchings necessary to cover the edge‐set of a cubic bridgeless graph is strictly related to a famous conjecture of Berge and Fulkerson. In this article, we prove that deciding whether this number is at most four for a given cubic bridgeless graph is NP‐complete. We also construct an infinite family of snarks (cyclically 4‐edge‐connected cubic graphs of girth at least 5 and chromatic index 4) whose edge‐set cannot be covered by four perfect matchings. Only two such graphs were known. It turns out that the family also has interesting properties with respect to the shortest cycle cover problem. The shortest cycle cover of any cubic bridgeless graph with m edges has length at least , and we show that this inequality is strict for graphs of . We also construct the first known snark with no cycle cover of length less than .     

Subgraphs smaller than the girth

For graphs A, B, let () denote the number of subsets of nodes of A for which the induced subgraph is B. If G and H both have girth > k, and if () = () for every k-node tree T, then for every k-node forest F, () = (). Say the spread of a tree is the number of nodes in a longest path. If G is regular of degree d, on n nodes, with girth > k, and if F is a forest of total spread ≤k, then the value of () depends only on n and d.     

A note on shortest cycle covers of cubic graphs

Let SCC₃(G) be the length of a shortest 3‐cycle cover of a bridgeless cubic graph G. It is proved in this note that if G contains no circuit of length 5 (an improvement of Jackson's (JCTB 1994) result: if G has girth at least 7) and if all 5‐circuits of G are disjoint (a new upper bound of SCC₃(G) for the special class of graphs).     

1‐Factor and Cycle Covers of Cubic Graphs

Let G be a bridgeless cubic graph. Consider a list of k 1‐factors of G. Let be the set of edges contained in precisely i members of the k 1‐factors. Let be the smallest over all lists of k 1‐factors of G. Any list of three 1‐factors induces a core of a cubic graph. We use results on the structure of cores to prove sufficient conditions for Berge‐covers and for the existence of three 1‐factors with empty intersection. Furthermore, if , then is an upper bound for the girth of G. We also prove some new upper bounds for the length of shortest cycle covers of bridgeless cubic graphs. Cubic graphs with have a 4‐cycle cover of length and a 5‐cycle double cover. These graphs also satisfy two conjectures of Zhang 18 . We also give a negative answer to a problem stated in 18 .