Minimum k-hamiltonian graphs,II

We consider in this paper graphs which remain hamiltonian after the removal of k edges (k-edge hamiltonian) or k vertices (k-hamiltonian). These classes of graphs arise from reliability considerations in network design. In a previous paper, W. W. Wong and C. K. Wong presented families of minimum k-hamiltonian graphs and minimum k-edge hamiltonian graphs for k even. Here, we complete this study in the case where k is odd.     

Powers of Hamiltonian cycles in randomly augmented graphs

We study the existence of powers of Hamiltonian cycles in graphs with large minimum degree to which some additional edges have been added in a random manner. It follows from the theorems of Dirac and of Komlós, Sarközy, and Szemerédi that for every k ≥ 1 and sufficiently large n already the minimum degree for an n‐vertex graph G alone suffices to ensure the existence of a kth power of a Hamiltonian cycle. Here we show that under essentially the same degree assumption the addition of just O(n) random edges ensures the presence of the (k + 1)st power of a Hamiltonian cycle with probability close to one.     

The graphical complexity of direct products of permutation groups

In this article, we improve known results, and, with one exceptional case, prove that when k≥3, the direct product of the automorphism groups of graphs whose edges are colored using k colors, is itself the automorphism group of a graph whose edges are colored using k colors. We have handled the case k = 2 in an earlier article. We prove similar results for directed edge‐colored graphs. © 2010 Wiley Periodicals, Inc. J Graph Theory 66:303‐318, 2011     

The Patterson–Sullivan Embedding and Minimal Volume Entropy for Outer Space

Motivated by Bonahon’s result for hyperbolic surfaces, we construct an analogue of the Patterson–Sullivan–Bowen–Margulis map from the Culler–Vogtmann outer space CV (F _k ) into the space of projectivized geodesic currents on a free group. We prove that this map is a continuous embedding and thus obtain a new compactification of the outer space. We also prove that for every k ≥ 2 the minimum of the volume entropy of the universal covers of finite connected volume-one metric graphs with fundamental group of rank k and without degree-one vertices is equal to (3k − 3) log 2 and that this minimum is realized by trivalent graphs with all edges of equal lengths, and only by such graphs. Received: December 2005, Accepted: March 2006     

The Erd s-Sós conjecture for graphs of girth 5

We prove that every graph of girth at least 5 with minimum degree δ k/2 contains every tree with k edges, whose maximum degree does not exceed the maximum degree of the graph. An immediate consequence is that the famous Erd s-Sós Conjecture, saying that every graph of order n with more than n(k − 1)/2 edges contains every tree with k edges, is true for graphs of girth at least 5.     

Closure and stable Hamiltonian properties in claw‐free graphs

In the class of k‐connected claw‐free graphs, we study the stability of some Hamiltonian properties under a closure operation introduced by the third author. We prove that (i) the properties of pancyclicity, vertex pancyclicity and cycle extendability are not stable for any k (i.e., for any of these properties there is an infinite family of graphs G_k of arbitrarily high connectivity k such that the closure of G_k has the property while the graph G_k does not); (ii) traceability is a stable property even for k = 1; (iii) homogeneous traceability is not stable for k = 2 (although it is stable for k = 7). The article is concluded with several open questions concerning stability of homogeneous traceability and Hamiltonian connectedness. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 30–41, 2000     

Algorithms for coloring semi-random graphs

The graph coloring problem is to color a given graph with the minimum number of colors. This problem is known to be NP-hard even if we are only aiming at approximate solutions. On the other hand, the best known approximation algorithms require n^δ (δ>0) colors even for bounded chromatic (k-colorable for fixed k) n-vertex graphs. The situation changes dramatically if we look at the average performance of an algorithm rather than its worst case performance. A k-colorable graph drawn from certain classes of distributions can be k-colored almost surely in polynomial time. It is also possible to k-color such random graphs in polynomial average time. In this paper, we present polynomial time algorithms for k-coloring graphs drawn from the semirandom model. In this model, the graph is supplied by an adversary each of whose decisions regarding inclusion of edges is reversed with some probability p. In terms of randomness, this model lies between the worst case model and the usual random model where each edge is chosen with equal probability. We present polynomial time algorithms of two different types. The first type of algorithms always run in polynomial time and succeed almost surely. Blum and Spencer [J. Algorithms, 19 , 204–234 (1995)] have also obtained independently such algorithms, but our results are based on different proof techniques which are interesting in their own right. The second type of algorithms always succeed and have polynomial running time on the average. Such algorithms are more useful and more difficult to obtain than the first type of algorithms. Our algorithms work for semirandom graphs drawn from a wide range of distributions and work as long as p≥n^−α(k)+ϵ where α(k)=(2k)/((k−1)(k+2)) and ϵ is a positive constant. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 13, 125–158 (1998)     

Bicolored Matchings in Some Classes of Graphs

We consider the problem of finding in a graph a set R of edges to be colored in red so that there are maximum matchings having some prescribed numbers of red edges. For regular bipartite graphs with n nodes on each side, we give sufficient conditions for the existence of a set R with |R|=n+1 such that perfect matchings with k red edges exist for all k,0≤k≤n. Given two integers p<q we also determine the minimum cardinality of a set R of red edges such that there are perfect matchings with p red edges and with q red edges. For 3-regular bipartite graphs, we show that if p≤4 there is a set R with |R|=p for which perfect matchings M_k exist with |M_k∩R|≤k for all k≤p. For trees we design a linear time algorithm to determine a minimum set R of red edges such that there exist maximum matchings with k red edges for the largest possible number of values of k.     

Game chromatic index of k‐degenerate graphs

We consider the following edge coloring game on a graph G. Given t distinct colors, two players Alice and Bob, with Alice moving first, alternately select an uncolored edge e of G and assign it a color different from the colors of edges adjacent to e. Bob wins if, at any stage of the game, there is an uncolored edge adjacent to colored edges in all t colors; otherwise Alice wins. Note that when Alice wins, all edges of G are properly colored. The game chromatic index of a graph G is the minimum number of colors for which Alice has a winning strategy. In this paper, we study the edge coloring game on k‐degenerate graphs. We prove that the game chromatic index of a k‐degenerate graph is at most Δ + 3k − 1, where Δ is the maximum vertex degree of the graph. We also show that the game chromatic index of a forest of maximum degree 3 is at most 4 when the forest contains an odd number of edges. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 144–155, 2001     

On the Number of Edges in Colour-Critical Graphs and Hypergraphs

A (hyper)graph G is called k-critical if it has chromatic number k, but every proper sub(hyper)graph of it is (k-1)-colourable. We prove that for sufficiently large k, every k-critical triangle-free graph on n vertices has at least (k-o(k))n edges. Furthermore, we show that every (k+1)-critical hypergraph on n vertices and without graph edges has at least (k-3/³?{k}) n(k-3/\sqrt[3]{k}) n edges. Both bounds differ from the best possible bounds by o(kn) even for graphs or hypergraphs of arbitrary girth.     

Lower bound of the hadwiger number of graphs by their average degree

The aim of this paper is to show that the minimum Hadwiger number of graphs with average degreek isO(k/√logk). Specially, it follows that Hadwiger’s conjecture is true for almost all graphs withn vertices, furthermore ifk is large enough then for almost all graphs withn vertices andnk edges.     

Primitive non-powerful symmetric loop-free signed digraphs with given base and minimum number of arcs

In [B.M. Kim, B.C. Song, W. Hwang, Primitive graphs with given exponents and minimum number of edges, Linear Algebra Appl. 420 (2007) 648-662], the minimum number of edges of a simple graph on n vertices with exponent k was determined. In this paper, we completely determine the minimum number, H(n,k), of arcs of primitive non-powerful symmetric loop-free signed digraphs on n vertices with base k, characterize the underlying digraphs which have H(n,k) arcs when k is 2, nearly characterize the case when k is 3 and propose an open problem.     

Codes from incidence matrices of graphs

We examine the p-ary codes, for any prime p, from the row span over ${\mathbb {F}_p}$ of |V| × |E| incidence matrices of connected graphs Γ = (V, E), showing that certain properties of the codes can be directly derived from the parameters and properties of the graphs. Using the edge-connectivity of Γ (defined as the minimum number of edges whose removal renders Γ disconnected) we show that, subject to various conditions, the codes from such matrices for a wide range of classes of connected graphs have the property of having dimension |V| or |V| ? 1, minimum weight the minimum degree δ(Γ), and the minimum words the scalar multiples of the rows of the incidence matrix of this weight. We also show that, in the k-regular case, there is a gap in the weight enumerator between k and 2k ? 2 of the binary code, and also for the p-ary code, for any prime p, if Γ is bipartite. We examine also the implications for the binary codes from adjacency matrices of line graphs. Finally we show that the codes of many of these classes of graphs can be used for permutation decoding for full error correction with any information set.     

Algorithms for finding distance-edge-colorings of graphs

For a bounded integer ℓ, we wish to color all edges of a graph G so that any two edges within distance ℓ have different colors. Such a coloring is called a distance-edge-coloring or an ℓ-edge-coloring of G. The distance-edge-coloring problem is to compute the minimum number of colors required for a distance-edge-coloring of a given graph G. A partial k-tree is a graph with tree-width bounded by a fixed constant k. We first present a polynomial-time exact algorithm to solve the problem for partial k-trees, and then give a polynomial-time 2-approximation algorithm for planar graphs.     

On the Size‐Ramsey Number of Hypergraphs

The size‐Ramsey number of a graph G is the minimum number of edges in a graph H such that every 2‐edge‐coloring of H yields a monochromatic copy of G. Size‐Ramsey numbers of graphs have been studied for almost 40 years with particular focus on the case of trees and bounded degree graphs. We initiate the study of size‐Ramsey numbers for k‐uniform hypergraphs. Analogous to the graph case, we consider the size‐Ramsey number of cliques, paths, trees, and bounded degree hypergraphs. Our results suggest that size‐Ramsey numbers for hypergraphs are extremely difficult to determine, and many open problems remain.     

Gallai–Ramsey numbers for cycles

For given graphs G and H and an integer k, the Gallai–Ramsey number is defined to be the minimum integer n such that, in any k coloring of the edges of K_n, there exists a subgraph isomorphic to either a rainbow coloring of G or a monochromatic coloring of H. In this work, we consider Gallai–Ramsey numbers for the case when G=K₃ and H is a cycle of a fixed length.     

Forcing highly connected subgraphs

A theorem of Mader states that highly connected subgraphs can be forced in finite graphs by assuming a high minimum degree. We extend this result to infinite graphs. Here, it is necessary to require not only high degree for the vertices but also high vertex‐degree (or multiplicity) for the ends of the graph, that is, a large number of disjoint rays in each end. We give a lower bound on the degree of vertices and the vertex‐degree of the ends which is quadratic in k, the connectedness of the desired subgraph. In fact, this is not far from best possible: we exhibit a family of graphs with a degree of order 2^k at the vertices and a vertex‐degree of order k log k at the ends which have no k‐connected subgraphs. Furthermore, if in addition to the high degrees at the vertices, we only require high edge‐degree for the ends (which is defined as the maximum number of edge‐disjoint rays in an end), Mader's theorem does not extend to infinite graphs, not even to locally finite ones. We give a counterexample in this respect. But, assuming a lower bound of at least 2k for the edge‐degree at the ends and the degree at the vertices does suffice to ensure the existence (k + 1)‐edge‐connected subgraphs in arbitrary graphs. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 331–349, 2007     

Special subdivisions ofK 4 and 4-chromatic graphs

In this paper we consider special subdivisions ofK ₄ in which some of the edges are left undivided. A best possible extremal-result for the case where the edges of a Hamiltonian path are left undivided is obtained. Moreover special subdivisions as subgraphs of 4-chromatic graphs are studied. Our main-result on 4-chromatic graphs says that any 4-critical graphG contains an odd cycleC without diagonals such thatG-V (C) is connected.     

Some properties of k-Delaunay and k-Gabriel graphs

We consider two classes of higher order proximity graphs defined on a set of points in the plane, namely, the k-Delaunay graph and the k-Gabriel graph. We give bounds on the following combinatorial and geometric properties of these graphs: spanning ratio, diameter, connectivity, chromatic number, and minimum number of layers necessary to partition the edges of the graphs so that no two edges of the same layer cross.     

One-diregular subgraphs in semicomplete multipartite digraphs

The problem of finding necessary and sufficient conditions for a semicomplete multipartite digraph (SMD) to be Hamiltonian, seems to be both very interesting and difficult. Bang-Jensen, Gutin and Huang ( Discrete Math to appear) proved a sufficient condition for a SMD to be Hamiltonian. A strengthening of this condition, shown in this paper, allows us to prove the following three results. We prove that every k-strong SMD with at most k-vertices in each color class is Hamiltonian and every k-strong SMD has a cycle through any set of k vertices. These two statements were stated as conjectures by Volkmann (L. Volkmann, a talk at the second Krakw Conference of Graph Theory (1994)) and Bang-Jensen, Gutin, and Yeo (J. Bang-Jensen, G. Gutin, and A. Yeo, On k-strong and k-cyclic digraphs, submitted), respectively. We also prove that every diregular SMD is Hamiltonian, which was conjectured in a weaker form by Zhang (C.-Q. Zhang, Hamilton paths in multipartite oriented graphs, Ann Discrete Math. 41 (1989), 499–581). © 1997 John Wiley & Sons, Inc.