Claw‐decompositions and tutte‐orientations

We conjecture that, for each tree T, there exists a natural number k_T such that the following holds: If G is a k_T‐edge‐connected graph such that |E(T)| divides |E(G)|, then the edges of G can be divided into parts, each of which is isomorphic to T. We prove that for T = K_1,3 (the claw), this holds if and only if there exists a (smallest) natural number k_t such that every k_t‐edge‐connected graph has an orientation for which the indegree of each vertex equals its outdegree modulo 3. Tutte's 3‐flow conjecture says that k_t = 4. We prove the weaker statement that every 4$\lceil$ log n$\rceil$ ‐edge‐connected graph with n vertices has an edge‐decomposition into claws provided its number of edges is divisible by 3. We also prove that every triangulation of a surface has an edge‐decomposition into claws. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 135–146, 2006     

Cycle covers of graphs with a nowhere-zero 4-flow

It is shown that the edges of a simple graph with a nowhere-zero 4-flow can be covered with cycles such that the sum of the lengths of the cycles is at most |E(G)| + |V(G)| ?3. This solves a conjecture proposed by G. Fan.     

Connected subgraphs with small degree sums in 3‐connected planar graphs

It is well‐known that every planar graph has a vertex of degree at most five. Kotzig proved that every 3‐connected planar graph has an edge xy such that deg(x) + deg (y) ≤ 13. In this article, considering a similar problem for the case of three or more vertices that induce a connected subgraph, we show that, for a given positive integer t, every 3‐connected planar graph G with |V(G)| ≥ t has a connected subgraph H of order t such that Σ_x∈V(H) deg_G(x) ≤ 8t − 1. As a tool for proving this result, we consider decompositions of 3‐connected planar graphs into connected subgraphs of order at least t and at most 2t − 1. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 191–203, 1999     

On packing 3‐vertex paths in a graph

Let G be a connected graph and let eb(G) and λ(G) denote the number of end‐blocks and the maximum number of disjoint 3‐vertex paths Λ in G. We prove the following theorems on claw‐free graphs: (t1) if G is claw‐free and eb(G) ≤ 2 (and in particular, G is 2‐connected) then λ(G) = ⌊| V(G)|/3⌋; (t2) if G is claw‐free and eb(G) ≥ 2 then λ(G) ≥ ⌊(| V(G) | − eb(G) + 2)/3 ⌋; and (t3) if G is claw‐free and Δ^*‐free then λ(G) = ⌊| V(G) |/3⌋ (here Δ^* is a graph obtained from a triangle Δ by attaching to each vertex a new dangling edge). We also give the following sufficient condition for a graph to have a Λ‐factor: Let n and p be integers, 1 ≤ p ≤ n − 2, G a 2‐connected graph, and |V(G)| = 3n. Suppose that G − S has a Λ‐factor for every S ⊆ V(G) such that |S| = 3p and both V(G) − S and S induce connected subgraphs in G. Then G has a Λ‐factor. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 175–197, 2001     

On a conjecture of wang and williams

Wang and Williams defined a threshold assignment for a graph G as an assignment of a non-negative weight to each vertex and edge of G, and a threshold t, such that a set S of vertices is stable if and only if the total weight of the subgraph induced by S does not exceed t - 1. This is always possible with zero vertex-weights, unit edge-weights, and t = 1. By definition, a threshold graph is a graph having a threshold assignment with zero edge-weights. The threshold weight of G is the minimum total edgeweight of a threshold assignment of G. A graph G = (V,E) is called heavy if its threshold weight is |E|. A clique C of G is called big if |E(C)| > |E(G - C)|. Wang and Williams showed that graphs with a big clique are not heavy, and conjectured the converse. They verified the conjecture for triangle-free graphs. We disprove the conjecture in general, but prove it for perfect graphs.     

Competitively Tight Graphs

The competition graph of a digraph D is a (simple undirected) graph which has the same vertex set as D and has an edge between two distinct vertices x and y if and only if there exists a vertex v in D such that (x, v) and (y, v) are arcs of D. For any graph G, G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number k(G) of a graph G is the smallest number of such isolated vertices. Computing the competition number of a graph is an NP-hard problem in general and has been one of the important research problems in the study of competition graphs. Opsut [1982] showed that the competition number of a graph G is related to the edge clique cover number θ _E (G) of the graph G via θ _E (G) ? |V(G)| + 2 ≤ k(G) ≤ θ _E (G). We first show that for any positive integer m satisfying 2 ≤ m ≤ |V(G)|, there exists a graph G with k(G) = θ _E (G) ? |V(G)| + m and characterize a graph G satisfying k(G) = θ _E (G). We then focus on what we call competitively tight graphs G which satisfy the lower bound, i.e., k(G) = θ _E (G) ? |V(G)| + 2. We completely characterize the competitively tight graphs having at most two triangles. In addition, we provide a new upper bound for the competition number of a graph from which we derive a sufficient condition and a necessary condition for a graph to be competitively tight.     

Total domination in graphs with minimum degree three

A set S of vertices of a graph G is a total dominating set, if every vertex of V(G) is adjacent to some vertex in S. The total domination number of G, denoted by γ_t(G), is the minimum cardinality of a total dominating set of G. We prove that, if G is a graph of order n with minimum degree at least 3, then γ_t(G) ≤ 7n/13. © 2000 John Wiley & Sons, Inc. J Graph Theory 34:9–19, 2000     

Finding minimum dominating cycles in permutation graphs

A dominating cycle for a graph G = (V, E) is a subset C of V which has the following properties: (i) the subgraph of G induced by C has a Hamiltonian cycle, and (ii) every vertex of V is adjacent to some vertex of C. In this paper, we develop an O(n²) algorithm for finding a minimum cardinality dominating cycle in a permutation graph. We also show that a minimum cardinality dominating cycle in a permutation graph always has an even number of vertices unless it is isomorphic to C₃.     

Some Notes on Signed Edge Domination in Graphs

The closed neighborhood N_G[e] of an edge e in a graph G is the set consisting of e and of all edges having an end-vertex in common with e. Let f be a function on E(G), the edge set of G, into the set {−1, 1}. If for each e ∈ E(G), then f is called a signed edge dominating function of G. The signed edge domination number γ_s^′(G) of G is defined as . Recently, Xu proved that γ_s^′(G) ≥ |V(G)| − |E(G)| for all graphs G without isolated vertices. In this paper we first characterize all simple connected graphs G for which γ_s^′(G) = |V(G)| − |E(G)|. This answers Problem 4.2 of [4]. Then we classify all simple connected graphs G with precisely k cycles and γ_s^′(G) = 1 − k, 2 − k. A. Khodkar: Research supported by a Faculty Research Grant, University of West Georgia. Send offprint requests to: Abdollah Khodkar.     

Partitions of a graph into paths with prescribed endvertices and lengths

For a graph G, let σ₂(G) denote the minimum degree sum of a pair of nonadjacent vertices. We conjecture that if |V(G)| = n = Σ^k_{i = 1} a_i and σ₂(G) ≥ n + k − 1, then for any k vertices v₁, v₂,…, v_k in G, there exist vertex‐disjoint paths P₁, P₂,…, P_k such that |V(P_i)| = a_i and v_i is an endvertex of P_i for 1 ≤ i ≤ k. In this paper, we verify the conjecture for the cases where almost all a_i ≤ 5, and the cases where k ≤ 3. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 163–169, 2000     

An Upper Bound for the Total Domination Subdivision Number of a Graph

A set S of vertices of a graph G = (V, E) without isolated vertex is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. The total domination number γ _t (G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number sd_gt(G){{\rm sd}_{\gamma_t}(G)} is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the total domination number. In this paper, we prove that sd_gt(G) ￡ 2g_t(G)-1{{\rm sd}_{\gamma_t}(G)\leq 2\gamma_t(G)-1} for every simple connected graph G of order n ≥ 3.     

On equality in an upper bound for the restrained and total domination numbers of a graph

Let G=(V,E) be a graph. A set S⊆V is a restrained dominating set (RDS) if every vertex not in S is adjacent to a vertex in S and to a vertex in V?S. The restrained domination number of G, denoted by γ_r(G), is the minimum cardinality of an RDS of G. A set S⊆V is a total dominating set (TDS) if every vertex in V is adjacent to a vertex in S. The total domination number of a graph G without isolated vertices, denoted by γ_t(G), is the minimum cardinality of a TDS of G.Let δ and Δ denote the minimum and maximum degrees, respectively, in G. If G is a graph of order n with δ?2, then it is shown that γ_r(G)?n-Δ, and we characterize the connected graphs with δ?2 achieving this bound that have no 3-cycle as well as those connected graphs with δ?2 that have neither a 3-cycle nor a 5-cycle. Cockayne et al. [Total domination in graphs, Networks 10 (1980) 211-219] showed that if G is a connected graph of order n?3 and Δ?n-2, then γ_t(G)?n-Δ. We further characterize the connected graphs G of order n?3 with Δ?n-2 that have no 3-cycle and achieve γ_t(G)=n-Δ.     

Adjacent strong edge colorings and total colorings of regular graphs

It is conjectured that χas(G) = χt(G) for every k-regular graph G with no C5 component (k 2). This conjecture is shown to be true for many classes of graphs, including: graphs of type 1; 2-regular, 3-regular and (|V (G)| - 2)-regular graphs; bipartite graphs; balanced complete multipartite graphs; k-cubes; and joins of two matchings or cycles.     

Domination number in graphs with minimum degree two

A set D of vertices of a graph G = （V, E） is called a dominating set if every vertex of V not in D is adjacent to a vertex of D. In 1996, Reed proved that every graph of order n with minimum degree at least 3 has a dominating set of cardinality at most 3n/8. In this paper we generalize Reed＇s result. We show that every graph G of order n with minimum degree at least 2 has a dominating set of cardinality at most （3n ＋IV21）/8, where V2 denotes the set of vertices of degree 2 in G. As an application of the above result, we show that for k ≥ 1, the k-restricted domination number rk （G, γ） ≤ （3n＋5k）/8 for all graphs of order n with minimum degree at least 3.     

The signed star domination numbers of the Cartesian product graphs

Let G be a graph with vertex set V(G) and edge set E(G). A function f:E(G)→{-1,1} is said to be a signed star dominating function of G if for every v∈V(G), where E_G(v)={uv∈E(G)|u∈V(G)}. The minimum of the values of , taken over all signed star dominating functions f on G, is called the signed star domination number of G and is denoted by γ_SS(G). In this paper, a sharp upper bound of γ_SS(G×H) is presented.     

Balanced judicious bipartitions of graphs

A bipartition of the vertex set of a graph is called balanced if the sizes of the sets in the bipartition differ by at most one. B. Bollobás and A. D. Scott, Random Struct Alg 21 (2002), 414–430 conjectured that if G is a graph with minimum degree of at least 2 then V(G) admits a balanced bipartition V₁, V₂ such that for each i, G has at most |E(G)|/3 edges with both ends in V_i. The minimum degree condition is necessary, and a result of B. Bollobás and A. D. Scott, J. Graph Theory 46 (2004), 131–143 shows that this conjecture holds for regular graphs G(i.e., when Δ(G)=δ(G)). We prove this conjecture for graphs G with \begin{eqnarray*}\Delta(G)\le\frac{7}{5}\delta(G)\end{eqnarray*}; hence, it holds for graphs ]ensuremathG with \begin{eqnarray*}\delta(G)\ge\frac{5}{7}|V(G)|\end{eqnarray*}. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 210–225, 2010     

Dominating Sets of Some Graphs Associated to Commutative Rings

Let G = (V, E) be a graph. A set S ? V is a dominating set of G if every vertex in V is either in S or is adjacent to a vertex in S. The domination number γ(G) of G is the minimum cardinality among the dominating sets of G. The main object of this article is to study and characterize the dominating sets of the zero-divisor graph Γ(R) and ideal-based zero-divisor graph Γ _I (R) of a commutative ring R.     

Dominating sets in triangulations on surfaces

Let G be a graph and let S?V(G). We say that S is dominating in G if each vertex of G is in S or adjacent to a vertex in S. We show that every triangulation on the torus and the Klein bottle with n vertices has a dominating set of cardinality at most $\frac{n}{3}Let G be a graph and let S?V(G). We say that S is dominating in G if each vertex of G is in S or adjacent to a vertex in S. We show that every triangulation on the torus and the Klein bottle with n vertices has a dominating set of cardinality at most $\frac{n}{3}$. Moreover, we show that the same conclusion holds for a triangulation on any non‐spherical surface with sufficiently large representativity. These results generalize that for plane triangulations proved by Matheson and Tarjan (European J Combin 17 (1996), 565–568), and solve a conjecture by Plummer (Private Communication). © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 17–30, 2010     

Remarks on the second neighborhood problem

The second neighborhood conjecture of Seymour asserts that for any orientation G = (V,E), there exists a vertex υ ∈ V so that |N⁺(υ)| ≤ |N⁺⁺(υ)|. The conjecture was resolved by Fisher for tournaments. In this article, we prove the second neighborhood conjecture for several additional classes of dense orientations. We also prove some approximation results, and reduce an asymptotic version of the conjecture to a finite case. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 208–220, 2007     

Decomposing graphs into paths of fixed length

Barát and Thomassen have conjectured that, for any fixed tree T, there exists a natural number k _T such that the following holds: If G is a k _T -edge-connected graph such that |E(T)| divides |E(G)|, then G has a T-decomposition. The conjecture is trivial when T has one or two edges. Before submission of this paper, the conjecture had been verified only for two other trees: the paths of length 3 and 4, respectively. In this paper we verify the conjecture for each path whose length is a power of 2.