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.     

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.     

Factors and vertex-deleted subgraphs

A relationship is considered between an f-factor of a graph and that of its vertex-deleted subgraphs. Katerinis [Some results on the existence of 2n-factors in terms of vertex-deleted subgraphs, Ars Combin. 16 (1983) 271-277] proved that for even integer k, if G-x has a k-factor for each x∈V(G), then G has a k-factor. Enomoto and Tokuda [Complete-factors and f-factors, Discrete Math. 220 (2000) 239-242] generalized Katerinis’ result to f-factors, and proved that if G-x has an f-factor for each x∈V(G), then G has an f-factor for an integer-valued function f defined on V(G) with even. In this paper, we consider a similar problem to that of Enomoto and Tokuda, where for several vertices x we do not have to know whether G-x has an f-factor. Let G be a graph, X be a set of vertices, and let f be an integer-valued function defined on V(G) with even, |V(G)-X|?2. We prove that if and if G-x has an f-factor for each x∈V(G)-X, then G has an f-factor. Moreover, if G excludes an isolated vertex, then we can replace the condition with . Furthermore the condition will be when |X|=1.     

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.     

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.     

Total restrained domination in trees

Let G=(V,E) be a graph. A set S⊆V is a total restrained dominating set if every vertex is adjacent to a vertex in S and every vertex of V-S is adjacent to a vertex in V-S. The total restrained domination number of G, denoted by γ_tr(G), is the smallest cardinality of a total restrained dominating set of G. We show that if T is a tree of order n, then . Moreover, we show that if T is a tree of order , then . We then constructively characterize the extremal trees T of order n achieving these lower bounds.     

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, .     

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.     

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.     

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.     

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.     

On conjugate adjacency matrices of a graph

Let G be a simple graph of order n. Let and , where a and b are two nonzero integers and m is a positive integer such that m is not a perfect square. We say that A^c=[c_ij] is the conjugate adjacency matrix of the graph G if c_ij=c for any two adjacent vertices i and j, for any two nonadjacent vertices i and j, and c_ij=0 if i=j. Let P_G(λ)=|λI-A| and denote the characteristic polynomial and the conjugate characteristic polynomial of G, respectively. In this work we show that if then , where denotes the complement of G. In particular, we prove that if and only if P_G(λ)=P_H(λ) and . Further, let P^c(G) be the collection of conjugate characteristic polynomials of vertex-deleted subgraphs G_i=G?i(i=1,2,…,n). If P^c(G)=P^c(H) we prove that , provided that the order of G is greater than 2.     

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 .     

A note on edge-graceful spectra of the square of paths

For a simple path P_r on r vertices, the square of P_r is the graph on the same set of vertices of P_r, and where every pair of vertices of distance two or less in P_r is connected by an edge. Given a (p,q)-graph G with p vertices and q edges, and a nonnegative integer k, G is said to be k-edge-graceful if the edges can be labeled bijectively by k,k+1,…,k+q−1, so that the induced vertex sums are pairwise distinct, where the vertex sum at a vertex is the sum of the labels of all edges incident to such a vertex, modulo the number of vertices p. We call the set of all such k the edge-graceful spectrum of G, and denote it by egI(G). In this article, the edge-graceful spectrum for the square of paths is completely determined for odd r.     

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.     

Two classes of edge domination in graphs

Let (, resp.) be the number of (local) signed edge domination of a graph G [B. Xu, On signed edge domination numbers of graphs, Discrete Math. 239 (2001) 179-189]. In this paper, we prove mainly that and hold for any graph G of order n(n?4), and pose several open problems and conjectures.     

The tail is cut for Ramsey numbers of cubes

The Ramsey number R(G) of a graph G is the least integer p such that for all bicolorings of the edges of the complete graph K_p, one of the monochromatic subgraphs contains a copy of G. We show that for any positive constant c and bipartite graph G=(U,V;E) of order n where the maximum degree of vertices in U is at most , . Moreover, we show that the Ramsey number of the cube Q_n of dimension n satisfies . In both cases, the small terms are removed from the powers in the upper bounds of a earlier result of the author.     

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.     

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.