Minimally (k,k)‐edge‐connected graphs

For an integer l > 1, the l‐edge‐connectivity of a connected graph with at least l vertices is the smallest number of edges whose removal results in a graph with l components. A connected graph G is (k, l)‐edge‐connected if the l‐edge‐connectivity of G is at least k. In this paper, we present a structural characterization of minimally (k, k)‐edge‐connected graphs. As a result, former characterizations of minimally (2, 2)‐edge‐connected graphs in [J of Graph Theory 3 (1979), 15–22] are extended. © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 116–131, 2003     

Extremal graphs in connectivity augmentation

Let A(n, k, t) denote the smallest integer e for which every k‐connected graph on n vertices can be made (k + t)‐connected by adding e new edges. We determine A(n, k, t) for all values of n, k, and t in the case of (directed and undirected) edge‐connectivity and also for directed vertex‐connectivity. For undirected vertex‐connectivity we determine A(n, k, 1) for all values of n and k. We also describe the graphs that attain the extremal values. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 179–193, 1999     

Removable edges in a 5-connected graph and a construction method of 5-connected graphs

An edge e of a k-connected graph G is said to be a removable edge if G?e is still k-connected. A k-connected graph G is said to be a quasi (k+1)-connected if G has no nontrivial k-separator. The existence of removable edges of 3-connected and 4-connected graphs and some properties of quasi k-connected graphs have been investigated [D.A. Holton, B. Jackson, A. Saito, N.C. Wormale, Removable edges in 3-connected graphs, J. Graph Theory 14(4) (1990) 465-473; H. Jiang, J. Su, Minimum degree of minimally quasi (k+1)-connected graphs, J. Math. Study 35 (2002) 187-193; T. Politof, A. Satyanarayana, Minors of quasi 4-connected graphs, Discrete Math. 126 (1994) 245-256; T. Politof, A. Satyanarayana, The structure of quasi 4-connected graphs, Discrete Math. 161 (1996) 217-228; J. Su, The number of removable edges in 3-connected graphs, J. Combin. Theory Ser. B 75(1) (1999) 74-87; J. Yin, Removable edges and constructions of 4-connected graphs, J. Systems Sci. Math. Sci. 19(4) (1999) 434-438]. In this paper, we first investigate the relation between quasi connectivity and removable edges. Based on the relation, the existence of removable edges in k-connected graphs (k?5) is investigated. It is proved that a 5-connected graph has no removable edge if and only if it is isomorphic to K₆. For a k-connected graph G such that end vertices of any edge of G have at most k-3 common adjacent vertices, it is also proved that G has a removable edge. Consequently, a recursive construction method of 5-connected graphs is established, that is, any 5-connected graph can be obtained from K₆ by a number of θ⁺-operations. We conjecture that, if k is even, a k-connected graph G without removable edge is isomorphic to either K_k+1 or the graph H_k/2+1 obtained from K_k+2 by removing k/2+1 disjoint edges, and, if k is odd, G is isomorphic to K_k+1.     

2‐factors with prescribed and proscribed edges

Let G be a finite k‐edge‐connected simple graph. We consider when a set of independent edges can be extended to a 2‐factor such that this 2‐factor avoids a fixed set of independent edges. A complete characterization is provided in those cases, where this is feasible. © 2005 Wiley Periodicals, Inc. J Graph Theory 49: 48–58, 2005     

On decomposing a graph into nontrivial bonds

We show that for every 2‐connected bipartite graph which is not a multiple edge and which has no K₅‐minor there is an edge‐disjoint collection of nontrivial bonds (i.e., not stars) which partition the edges of the graph. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 109–127, 2000     

Vertex partitions of chordal graphs

A k‐tree is a chordal graph with no (k + 2)‐clique. An ?‐tree‐partition of a graph G is a vertex partition of G into ‘bags,’ such that contracting each bag to a single vertex gives an ?‐tree (after deleting loops and replacing parallel edges by a single edge). We prove that for all k ≥ ? ≥ 0, every k‐tree has an ?‐tree‐partition in which each bag induces a connected ‐tree. An analogous result is proved for oriented k‐trees. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 167–172, 2006     

The size of minimum 3‐trees: Cases 3 and 4 mod 6

A 3‐uniform hypergraph is called a minimum 3‐tree, if for any 3‐coloring of its vertex set there is a heterochromatic edge and the hypergraph has the minimum possible number of edges. Here we show that the number of edges in such 3‐tree is for any number of vertices n ≡ 3, 4 (mod 6). © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 157‐166, 1999     

Characterization of edge‐colored complete graphs with properly colored Hamilton paths

An edge‐colored graph H is properly colored if no two adjacent edges of H have the same color. In 1997, J. Bang‐Jensen and G. Gutin conjectured that an edge‐colored complete graph G has a properly colored Hamilton path if and only if G has a spanning subgraph consisting of a properly colored path C₀ and a (possibly empty) collection of properly colored cycles C₁,C₂,…, C_d such that provided . We prove this conjecture. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 333–346, 2006     

Hamiltonian chains in hypergraphs

A cyclic ordering of the vertices of a k‐uniform hypergraph is called a hamiltonian chain if any k consecutive vertices in the ordering form an edge. For k = 2 this is the same as a hamiltonian cycle. We consider several natural questions about the new notion. The main result is a Dirac‐type theorem that provides a sufficient condition for finding hamiltonian chains in k‐uniform hypergraphs with large (k − 1)‐minimal degree. If it is more than than the hypergraph contains a hamiltonian chain. © 1999 Wiley & Sons, Inc. J Graph Theory 30: 205–212, 1999     

Analysis of parallel algorithms for finding a maximal independent set in a random hypergraph

It is well known [9] that finding a maximal independent set in a graph is in class NC and [10] that finding a maximal independent set in a hypergraph with fixed dimension is in RNC. It is not known whether this latter problem remains in NC when the dimension is part of the input. We will study the problem when the problem instances are randomly chosen. It was shown in [6] that the expected running time of a simple parallel algorithm for finding the lexicographically first maximal independent set (Ifmis) in a random simple graph is logarithmic in the input size. In this paper, we will prove a generalization of this result. We show that if a random k-uniform hypergraph has vertex set {1, 2, …, n} and its edges are chosen independently with probability p from the set of (ⁿ_k) possible edges, then our algorithm finds the Ifmis in O( ) expected time. The hidden constant is independent of k, p. © 1996 John Wiley & Sons, Inc. Random Struct. Alg., 9 , 359–377 (1996)     

Disjoint Hamilton cycles in the random geometric graph

We consider the standard random geometric graph process in which n vertices are placed at random on the unit square and edges are sequentially added in increasing order of edge‐length. For fixed k?1, weprove that the first edge in the process that creates a k‐connected graph coincides a.a.s. with the first edge that causes the graph to contain k/2 pairwise edge‐disjoint Hamilton cycles (for even k), or (k?1)/2 Hamilton cycles plus one perfect matching, all of them pairwise edge‐disjoint (for odd k). This proves and extends a conjecture of Krivelevich and M ler. In the special case when k = 2, our result says that the first edge that makes the random geometric graph Hamiltonian is a.a.s. exactly the same one that gives 2‐connectivity, which answers a question of Penrose. (This result appeared in three independent preprints, one of which was a precursor to this article.) We prove our results with lengths measured using the ?_p norm for any p>1, and we also extend our result to higher dimensions. © 2011 Wiley Periodicals, Inc. J Graph Theory 68:299‐322, 2011     

Color critical hypergraphs with many edges

We show that for all k ≥ 3, r > l ≥ 2 there exists constant c = c(k, r, l) such that for large enough n there exists a k‐color‐critical r‐uniform hypergraph on less than n vertices, having more than cn^l edges, and having no l‐set of vertices occuring in more than one edge. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 56–74, 2006     

On Four-Connecting a Triconnected Graph

We consider the problem of finding a smallest set of edges whose addition four-connects a triconnected graph. This is a fundamental graph-theoretic problem that has applications in designing reliable networks and improving statistical database security. We present an O(n · α(m, n) + m)-time algorithm for four-connecting an undirected graph G that is triconnected by adding the smallest number of edges, where n and m are the number of vertices and edges in G, respectively, and α(m, n) is the inverse Ackermann function. This is the first polynomial time algorithm to solve this problem exactly.In deriving our algorithm, we present a new lower bound for the number of edges needed to four-connect a triconnected graph. The form of this lower bound is different from the form of the lower bound known for biconnectivity augmentation and triconnectivity augmentation. Our new lower bound applies for arbitrary k and gives a tighter lower bound than the one known earlier for the number of edges needed to k-connect a (k − 1)-connected graph. For k = 4, we show that this lower bound is tight by giving an efficient algorithm to find a set of edges whose size equals the new lower bound and whose addition four-connects the input triconnected graph.     

An algorithm for finding factorizations of complete graphs

We show how to find a decomposition of the edge set of the complete graph into regular factors where the degree and edge‐connectivity of each factor is prescribed. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 132–136, 2003     

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     

Sufficient conditions for graphs to be λ′‐optimal,super‐edge‐connected,and maximally edge‐connected

The restricted‐edge‐connectivity of a graph G, denoted by λ′(G), is defined as the minimum cardinality over all edge‐cuts S of G, where G‐S contains no isolated vertices. The graph G is called λ′‐optimal, if λ′(G) = ξ(G), where ξ(G) is the minimum edge‐degree in G. A graph is super‐edge‐connected, if every minimum edge‐cut consists of edges adjacent to a vertex of minimum degree. In this paper, we present sufficient conditions for arbitrary, triangle‐free, and bipartite graphs to be λ′‐optimal, as well as conditions depending on the clique number. These conditions imply super‐edge‐connectivity, if δ (G) ≥ 3, and the equality of edge‐connectivity and minimum degree. Different examples will show that these conditions are best possible and independent of other results in this area. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 228–246, 2005     

A list version of Dirac's theorem on the number of edges in colour‐critical graphs

One of the basic results in graph colouring is Brooks' theorem [R. L. Brooks, Proc Cambridge Phil Soc 37 ( 4 ) 194–197], which asserts that the chromatic number of every connected graph, that is not a complete graph or an odd cycle, does not exceed its maximum degree. As an extension of this result, Dirac [G. A. Dirac, Proc London Math Soc 7(3) ( 7 ) 161–195] proved that every k‐colour‐critical graph (k ≥ 4) on n ≥ k + 2 vertices has at least ½((k ? 1) n + k ? 3) edges. The aim of this paper is to prove a list version of Dirac's result and to extend it to hypergraphs. © 2002 Wiley Periodicals, Inc. J Graph Theory 39: 165–177, 2002; DOI 10.1002/jgt.998     

An application of the combinatorial Nullstellensatz to a graph labelling problem

An antimagic labelling of a graph G with m edges and n vertices is a bijection from the set of edges of G to the set of integers {1,…,m}, such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with that vertex. A graph is called antimagic if it admits an antimagic labelling. In N. Hartsfield and G. Ringle, Pearls in Graph Theory, Academic Press, Inc., Boston, 1990, Ringel has conjectured that every simple connected graph, other than K₂, is antimagic. In this article, we prove a special case of this conjecture. Namely, we prove that if G is a graph on n=p^k vertices, where p is an odd prime and k is a positive integer that admits a C_p‐factor, then it is antimagic. The case p=3 was proved in D. Hefetz, J Graph Theory 50 (2005), 263–272. Our main tool is the combinatorial Nullstellensatz [N. Alon, Combin Probab Comput 8(1–2) (1999), 7–29]. © 2009 Wiley Periodicals, Inc. J Graph Theory 65: 70–82, 2010.     

On total 9‐coloring planar graphs of maximum degree seven

Given a graph G, a total k‐coloring of G is a simultaneous coloring of the vertices and edges of G with at most k colors. If Δ(G) is the maximum degree of G, then no graph has a total Δ‐coloring, but Vizing conjectured that every graph has a total (Δ + 2)‐coloring. This Total Coloring Conjecture remains open even for planar graphs. This article proves one of the two remaining planar cases, showing that every planar (and projective) graph with Δ ≤ 7 has a total 9‐coloring by means of the discharging method. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 67–73, 1999