On edge domination numbers of graphs

Let and be the signed edge domination number and signed star domination number of G, respectively. We prove that holds for all graphs G without isolated vertices, where n=|V(G)|?4 and m=|E(G)|, and pose some problems and conjectures.     

Total minus domination in k-partite graphs

A function f defined on the vertices of a graph G=(V,E),f:V→{-1,0,1} is a total minus dominating function (TMDF) if the sum of its values over any open neighborhood is at least one. The weight of a TMDF is the sum of its function values over all vertices. The total minus domination number, denoted by , of G is the minimum weight of a TMDF on G. In this paper, a sharp lower bound on of k-partite graphs is given.     

Total domination and total domination subdivision number of a graph and its complement

A set S of vertices of a graph G=(V,E) with no isolated vertex is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. The total domination numberγ_t(G) is the minimum cardinality of a total dominating set of G. The total domination subdivision numbersd_γt(G) is the minimum number of edges that must be subdivided in order to increase the total domination number. We consider graphs of order n?4, minimum degree δ and maximum degree Δ. We prove that if each component of G and has order at least 3 and , then and if each component of G and has order at least 2 and at least one component of G and has order at least 3, then . We also give a result on stronger than a conjecture by Harary and Haynes.     

Edge covered critical multigraphs

Let G be a multigraph with edge set E(G). An edge coloring C of G is called an edge covered coloring, if each color appears at least once at each vertex v∈V(G). The maximum positive integer k such that G has a k edge covered coloring is called the edge covered chromatic index of G and is denoted by . A graph G is said to be of class if and otherwise of class. A pair of vertices {u,v} is said to be critical if . A graph G is said to be edge covered critical if it is of class and every edge with vertices in V(G) not belonging to E(G) is critical. Some properties about edge covered critical graphs are considered.     

Edge-recognizable domination numbers

For any undirected graph G, let be the collection of edge-deleted subgraphs. It is always possible to construct a graph H from so that . The general edge-reconstruction conjecture states that G and H must be isomorphic if they have at least four edges. A graphical invariant that must be identical for all graphs that can be constructed from the given collection is said to be edge-recognizable. Here we show that the domination number and many of its common variations are edge-recognizable.     

Remarks on the minus (signed) total domination in graphs

A function f:V(G)→{+1,0,-1} defined on the vertices of a graph G is a minus total dominating function if the sum of its function values over any open neighborhood is at least 1. The minus total domination number of G is the minimum weight of a minus total dominating function on G. By simply changing “{+1,0,-1}” in the above definition to “{+1,-1}”, we can define the signed total dominating function and the signed total domination number of G. In this paper we present a sharp lower bound on the signed total domination number for a k-partite graph, which results in a short proof of a result due to Kang et al. on the minus total domination number for a k-partite graph. We also give sharp lower bounds on and for triangle-free graphs and characterize the extremal graphs achieving these bounds.     

On NP-hardness of the clique partition—Independence number gap recognition and related problems

We show that for a graph G it is NP-hard to decide whether its independence number α(G) equals its clique partition number even when some minimum clique partition of G is given. This implies that any α(G)-upper bound provably better than is NP-hard to compute.To establish this result we use a reduction of the quasigroup completion problem (QCP, known to be NP-complete) to the maximum independent set problem. A QCP instance is satisfiable if and only if the independence number α(G) of the graph obtained within the reduction is equal to the number of holes h in the QCP instance. At the same time, the inequality always holds. Thus, QCP is satisfiable if and only if . Computing the Lovász number ?(G) we can detect QCP unsatisfiability at least when . In the other cases QCP reduces to gap recognition, with one minimum clique partition of G known.     

Edges not contained in triangles and the number of contractible edges in a 4-connected graph

Let G be a 4-connected graph, and let E_c(G) denote the set of 4-contractible edges of G and let denote the set of those edges of G which are not contained in a triangle. Under this notation, we show that if , then we have .     

Distance irredundance and connected domination numbers of a graph

Let k be a positive integer and G be a connected graph. This paper considers the relations among four graph theoretical parameters: the k-domination number γ_k(G), the connected k-domination number ; the k-independent domination number and the k-irredundance number ir_k(G). The authors prove that if an ir_k-set X is a k-independent set of G, then , and that for k?2, if ir_k(G)=1, if ir_k(G) is odd, and if ir_k(G) is even, which generalize some known results.     

r-Strong edge colorings of graphs

Let G be a graph and for any natural number r, denotes the minimum number of colors required for a proper edge coloring of G in which no two vertices with distance at most r are incident to edges colored with the same set of colors. In [Z. Zhang, L. Liu, J. Wang, Adjacent strong edge coloring of graphs, Appl. Math. Lett. 15 (2002) 623-626] it has been proved that for any tree T with at least three vertices, . Here we generalize this result and show that . Moreover, we show that if for any two vertices u and v with maximum degree d(u,v)?3, then . Also for any tree T with Δ(T)?3 we prove that . Finally, it is shown that for any graph G with no isolated edges, .     

Hamilton decompositions of directed cubes and products

Call a directed graph symmetric if it is obtained from an undirected graph G by replacing each edge of G by two directed edges, one in each direction. We will show that if G has a Hamilton decomposition with certain additional structure, then has a directed Hamilton decomposition. In particular, it will follow that the bidirected cubes for m?2 are decomposable into 2m+1 directed Hamilton cycles and that a product of cycles is decomposable into 2m+1 directed Hamilton cycles if n_i?3 and m?2.     

A contraction of square transversal designs

In this paper we show that if a square transversal design TD_λ[k;u], say D(=(P,B)), admits a class semiregular automorphism group G of order s, then we have a by matrix M with entries from G∪{0} satisfying , where , if i=j, and , otherwise. As an application of (*), we show that any symmetric TD₂[12;6] admits no nontrivial elation. We also obtain a result that gives us a restriction on the existence of elations of putative projective planes of composite order.     

Coloring the complements of intersection graphs of geometric figures

Let be the complement of the intersection graph G of a family of translations of a compact convex figure in Rⁿ. When n=2, we show that , where γ(G) is the size of the minimum dominating set of G. The bound on is sharp. For higher dimension we show that , for n?3. We also study the chromatic number of the complement of the intersection graph of homothetic copies of a fixed convex body in Rⁿ.     

On signed cycle domination in graphs

Let G=(V,E) be a graph, a function f:E→{−1,1} is said to be an signed cycle dominating function (SCDF) of G if ∑_e∈E(C)f(e)≥1 holds for any induced cycle C of G. The signed cycle domination number of G is defined as is an SCDF of G}. In this paper, we obtain bounds on , characterize all connected graphs G with , and determine the exact value of for some special classes of graphs G. In addition, we pose some open problems and conjectures.     

The hamiltonian index of a 2-connected graph

Let G be a graph. Then the hamiltonian index h(G) of G is the smallest number of iterations of line graph operator that yield a hamiltonian graph. In this paper we show that for every 2-connected simple graph G that is not isomorphic to the graph obtained from a dipole with three parallel edges by replacing every edge by a path of length l≥3. We also show that for any two 2-connected nonhamiltonian graphs G and with at least 74 vertices. The upper bounds are all sharp.     

Long paths with endpoints in given vertex-subsets of graphs

Let G=(V,E) be a connected graph of order n, t a real number with t?1 and M⊆V(G) with . In this paper, we study the problem of some long paths to maintain their one or two different endpoints in M. We obtain the following two results: (1) for any vertex v∈V(G), there exists a vertex u∈M and a path P with the two endpoints v and u to satisfy , , d_G(u)+1-t}; (2) there exists either a cycle C to cover all vertices of M or a path P with two different endpoints u₀ and u_p in M to satisfy , where .     

Lattices generated by join of strongly closed subgraphs in d-bounded distance-regular graphs

Let Γ be a d-bounded distance-regular graph with diameter d?3. Suppose that P(x) is a set of all strongly closed subgraphs containing x and that P(x,i) is a subset of P(x) consisting of all elements of P(x) with diameter i. Let L^′(x,i) be the set generated by all joins of the elements in P(x,i). By ordering L^′(x,i) by inclusion or reverse inclusion, L^′(x,i) is denoted by or . We prove that and are both finite atomic lattices, and give the conditions for them both being geometric lattices. We also give the eigenpolynomial of      

Generalization of transitive fraternal augmentations for directed graphs and its applications

Let be a directed graph. A transitive fraternal augmentation of is a directed graph with the same vertex set, including all the arcs of and such that for any vertices x,y,z,

1.

if and then or (fraternity);

2.

if and then (transitivity).

In this paper, we explore some generalization of the transitive fraternal augmentations for directed graphs and its applications. In particular, we show that the 2-coloring number col₂(G)≤O(∇₁(G)∇₀(G)²), where ∇_k(G) (k≥0) denotes the greatest reduced average density with depth k of a graph G; we give a constructive proof that ∇_k(G) bounds the distance (k+1)-coloring number col_k+1(G) with a function f(∇_k(G)). On the other hand, ∇_k(G)≤(col_2k+1(G))^2k+1. We also show that an inductive generalization of transitive fraternal augmentations can be used to study nonrepetitive colorings of graphs.     

On wreathed lexicographic products of graphs

This paper proves a necessary and sufficient condition for the endomorphism monoid of a lexicographic product G[H] of graphs G,H to be the wreath product of the monoids and . The paper also gives respective necessary and sufficient conditions for specialized cases such as for unretractive or triangle-free graphs G.