Bounds for the 2-domination number of toroidal grid graphs

A k-dominating set for a graph G(V, E) is a set of vertices D? V such that every vertex v∈V\ D is adjacent to at least k vertices in D. The k-domination number of G, denoted by γ _k (G), is the cardinality of a smallest k-dominating set of G. Here we establish lower and upper bounds of γ _k (C _m ×C _n ) for k=2. In some cases, these bounds agree so that the exact 2-domination number is obtained.     

Independence and k-domination in graphs

A subset S of vertices of a graph G is k-dominating if every vertex not in S has at least k neighbours in S. The k-domination number γ _k (G) is the minimum cardinality of a k-dominating set of G, and α(G) denotes the cardinality of a maximum independent set of G. Brook's well-known bound for the chromatic number χ and the inequality α(G)≥n(G)/χ(G) for a graph G imply that α(G)≥n(G)/Δ(G) when G is non-regular and α(G)≥n(G)/(Δ(G)+1) otherwise. In this paper, we present a new proof of this property and derive some bounds on γ _k (G). In particular, we show that, if G is connected with δ(G)≥k then γ _k (G)≤(Δ(G)?1)α(G) with the exception of G being a cycle of odd length or the complete graph of order k+1. Finally, we characterize the connected non-regular graphs G satisfying equality in these bounds and present a conjecture for the regular case.     

On the k-tuple domination of generalized de Brujin and Kautz digraphs

A vertex u in a digraph G = (V, A) is said to dominate itself and vertices v such that (u, v) ∈ A. For a positive integer k, a k-tuple dominating set of G is a subset D of vertices such that every vertex in G is dominated by at least k vertices in D. The k-tuple domination number of G is the minimum cardinality of a k-tuple dominating set of G. This paper deals with the k-tuple domination problem on generalized de Bruijn and Kautz digraphs. We establish bounds on the k-tuple domination number for the generalized de Bruijn and Kautz digraphs and we obtain some conditions for the k-tuple domination number attaining the bounds.     

On the k-tuple domination of de Bruijn and Kautz digraphs

In a digraph G, a vertex u is said to dominate itself and vertices v such that (u,v) is an arc of G. For a positive integer k, a k-tuple dominating set D of a digraph is a subset of vertices such that every vertex is dominated by at least k vertices in D. The k-tuple domination number of a given digraph is the minimum cardinality of a k-tuple dominating set of the digraph. In this letter, we give the exact values of the k-tuple domination number of de Bruijn and Kautz digraphs.     

On the locatic number of graphs

A vertex v of a connected graph G distinguishes a pair u, w of vertices of G if d(v, u)≠d(v, w), where d(·,·) denotes the length of a shortest path between two vertices in G. A k-partition Π={S ₁, S ₂, …, S _k } of the vertex set of G is said to be a locatic partition if for every pair of distinct vertices v and w of G, there exists a vertex s∈S _i for all 1≤i≤k that distinguishes v and w. The cardinality of a largest locatic partition is called the locatic number of G. In this paper, we study the locatic number of paths, cycles and characterize all the connected graphs of order n having locatic number n, n?1 and n?2. Some realizable results are also given in this paper.     

Extremal values on the harmonic number of trees

Let G=(V(G), E(G)) be a simple connected graph. The harmonic number of G, denoted by H(G), is defined as the sum of the weights 2/(d(u)+d(v)) of all edges uv of G, where d(u) denotes the degree of a vertex u in G. In this paper, some extremal problems on the harmonic number of trees are studied. The extremal values on the harmonic number of trees with given graphic parameters, such as pendant number, matching number, domination number and diameter, are determined. The corresponding extremal graphs are characterized, respectively.     

The twin domination number in generalized de Bruijn digraphs

Let G=(V,A) be a digraph. A set T of vertices of G is a twin dominating set of G if for every vertex v∈V?T, there exist u,w∈T (possibly u=w) such that arcs (u,v),(v,w)∈A. The twin domination numberγ^∗(G) of G is the cardinality of a minimum twin dominating set of G. In this paper we investigate the twin domination number in generalized de Bruijn digraphs GB(n,d). For the digraphs GB(n,d), we first establish sharp bounds on the twin domination number. Secondly, we give the exact values of the twin domination number for several types of GB(n,d) by constructing minimum twin dominating sets in the digraphs. Finally, we present sharp upper bounds for some special generalized de Bruijn digraphs.     

Not necessarily proper total colourings which are adjacent vertex distinguishing

Let G be a simple graph with no isolated edge. Let f be a total colouring of G which is not necessarily proper. f is said to be adjacent vertex distinguishing if for any pair of adjacent vertices u, v of G, we have C(u)≠C(v), where C(u) denotes the set of colours of u and its incident edges under f. The minimum number of colours required for an adjacent vertex distinguishing not necessarily proper total colouring of G is called the adjacent vertex distinguishing not necessarily proper total chromatic number. Seven kinds of adjacent vertex distinguishing not necessarily proper total colourings are introduced in this paper. Some results of adjacent vertex distinguishing not necessarily proper total chromatic numbers are obtained and some conjectures are also proposed.     

Upper signed k-domination in a general graph

Let k be a positive integer, and let G=(V,E) be a graph with minimum degree at least k−1. A function f:V→{−1,1} is said to be a signed k-dominating function (SkDF) if _∑u∈N[v]f(u)?k for every v∈V. An SkDF f of a graph G is minimal if there exists no SkDF g such that g≠f and g(v)?f(v) for every v∈V. The maximum of the values of _∑v∈Vf(v), taken over all minimal SkDFs f, is called the upper signed k-domination numberΓkS(G). In this paper, we present a sharp upper bound on this number for a general graph.     

Maximal matching and edge domination in complete multipartite graphs

For any graph G, let α′(G) and α′_min(G) be the maximum cardinality and minimum cardinality among all maximal matchings in G, respectively, and let γ′(G) and γ_t ′(G) be the edge domination number and edge total domination number of G, respectively. In this paper, we first show some properties of maximal matchings and further determine the exact values of α′(G) and α′_min(G) for a complete multipartite graph G. Then, we disclose relationships between maximal matchings and minimal edge dominating sets, and thus obtain the exact values of γ′(G) and γ_t′(G) for a complete multipartite graph G.     

Radio number of ladder graphs

Let G be a connected graph with diameter diam(G). The radio number for G, denoted by rn(G), is the smallest integer k such that there exists a function f: V(G)→{0, 1, 2, …, k} with the following satisfied for all vertices u and v:|f(u)?f(v)|≥diam (G)?d _G (u, v)+1, where d _G (u, v) is the distance between u and v in G. In this paper, we determine the radio number of ladder graphs.     

On circular-L(2, 1)-labellings of products of graphs

Let m, j and k be positive integers. An m-circular-L(j, k)-labelling of a graph G is an assignment f from { 0, 1,?…?, m?1} to the vertices of G such that, for any two vertices u and v, |f(u)?f(v)|_m≥j if uv∈E(G), and |f(u)?f(v)|_m≥k if d_G(u, v)=2, where |a|_m=min{a, m?a}. The minimum m such that G has an m-circular-L(j, k)-labelling is called the circular-L(j, k)-labelling number of G. This paper determines the circular-L(2, 1)-labelling numbers of the direct product of a path and a complete graph and of the Cartesian product of a path and a cycle.     

On the 1-fault hamiltonicity for graphs satisfying Ore's theorem and its generalization

Consider any undirected and simple graph G=(V, E), where V and E denote the vertex set and the edge set of G, respectively. Let |G|=|V|=n. The well-known Ore's theorem states that if deg_G(u)+deg_G(v)≥n+k holds for each pair of nonadjacent vertices u and v of G, then G is traceable for k=?1, hamiltonian for k=0, and hamiltonian-connected for k=1. Lin et al. generalized Ore's theorem and showed that under the same condition as above, G is r*-connected for 1≤r≤k+2 with k≥1. In this paper, we improve both theorems by showing that the hamiltonicity or r*-connectivity of any graph G satisfying the condition deg_G(u)+deg_G(v)≥n+k with k≥?1 is preserved even after one vertex or one edge is removed, unless G belongs to two exceptional families.     

The super laceability of the hypercubes

A k-containerC(u,v) of a graph G is a set of k disjoint paths joining u to v. A k-container C(u,v) is a k∗-container if every vertex of G is incident with a path in C(u,v). A bipartite graph G is k∗-laceable if there exists a k∗-container between any two vertices u, v from different partite set of G. A bipartite graph G with connectivity k is super laceable if it is i∗-laceable for all i?k. A bipartite graph G with connectivity k is f-edge fault-tolerant super laceable if G−F is i∗-laceable for any 1?i?k−f and for any edge subset F with |F|=f<k−1. In this paper, we prove that the hypercube graph Q_r is super laceable. Moreover, Q_r is f-edge fault-tolerant super laceable for any f?r−2.     

Local colourings of Cartesian product graphs

A local colouring of a graph G is a function c: V(G)→? such that for each S ? V(G), 2≤|S|≤3, there exist u, v∈S with |c(u)?c(v)| at least the number of edges in the subgraph induced by S. The maximum colour assigned by c is the value χ_?(c) of c, and the local chromatic number of G is χ_?(G)=min {χ_?(c): c is a local colouring of G}. In this note the local chromatic number is determined for Cartesian products G □ H, where G and GH are 3-colourable graphs. This result in part corrects an error from Omoomi and Pourmiri [On the local colourings of graphs, Ars Combin. 86 (2008), pp. 147–159]. It is also proved that if G and H are graphs such that χ(G)≤? χ_?(H)/2 ?, then χ_?(G □ H)≤χ_?(H)+1.     

The super-connected property of recursive circulant graphs

In a graph G, a k-container C_k(u,v) is a set of k disjoint paths joining u and v. A k-container C_k(u,v) is k∗-container if every vertex of G is passed by some path in C_k(u,v). A graph G is k∗-connected if there exists a k∗-container between any two vertices. An m-regular graph G is super-connected if G is k∗-connected for any k with 1?k?m. In this paper, we prove that the recursive circulant graphs G(2^m,4), proposed by Park and Chwa [Theoret. Comput. Sci. 244 (2000) 35-62], are super-connected if and only if m≠2.     

Lucky labelings of graphs

Suppose the vertices of a graph G were labeled arbitrarily by positive integers, and let S(v) denote the sum of labels over all neighbors of vertex v. A labeling is lucky if the function S is a proper coloring of G, that is, if we have S(u)≠S(v) whenever u and v are adjacent. The least integer k for which a graph G has a lucky labeling from the set {1,2,…,k} is the lucky number of G, denoted by η(G).Using algebraic methods we prove that η(G)?k+1 for every bipartite graph G whose edges can be oriented so that the maximum out-degree of a vertex is at most k. In particular, we get that η(T)?2 for every tree T, and η(G)?3 for every bipartite planar graph G. By another technique we get a bound for the lucky number in terms of the acyclic chromatic number. This gives in particular that for every planar graph G. Nevertheless we offer a provocative conjecture that η(G)?χ(G) for every graph G.     

On edge connectivity of direct products of graphs

Let λ(G) be the edge connectivity of G. The direct product of graphs G and H is the graph with vertex set V(G×H)=V(G)×V(H), where two vertices (u₁,v₁) and (u₂,v₂) are adjacent in G×H if u₁u₂∈E(G) and v₁v₂∈E(H). We prove that λ(G×Kn)=min{n(n−1)λ(G),(n−1)δ(G)} for every nontrivial graph G and n?3. We also prove that for almost every pair of graphs G and H with n vertices and edge probability p, G×H is k-connected, where k=O(²(n/logn)).     

Constructing Labeling Schemes through Universal Matrices

Let f be a function on pairs of vertices. An f -labeling scheme for a family of graphs ℱ labels the vertices of all graphs in ℱ such that for every graph G∈ℱ and every two vertices u,v∈G, f(u,v) can be inferred by merely inspecting the labels of u and v. The size of a labeling scheme is the maximum number of bits used in a label of any vertex in any graph in ℱ. This paper illustrates that the notion of universal matrices can be used to efficiently construct f-labeling schemes.