Signed and minus clique-transversal functions on graphs

A minus (respectively, signed) clique-transversal function of a graph G=(V,E) is a function (respectively, {−1,1}) such that _∑u∈Cf(u)?1 for every maximal clique C of G. The weight of a minus (respectively, signed) clique-transversal function of G is f(V)=_∑v∈Vf(v). The minus (respectively, signed) clique-transversal problem is to find a minus (respectively, signed) clique-transversal function of G of minimum weight. In this paper, we present a unified approach to these two problems on strongly chordal graphs. Notice that trees, block graphs, interval graphs, and directed path graphs are subclasses of strongly chordal graphs. We also prove that the signed clique-transversal problem is NP-complete for chordal graphs and planar graphs.     

A Fan-type result on k-ordered graphs

For a positive integer k, a graph G is k-ordered hamiltonian if for every ordered sequence of k vertices there is a hamiltonian cycle that encounters the vertices of the sequence in the given order. In this paper, we show that if G is a ⌊3k/2⌋-connected graph of order n?100k, and d(u)+d(v)?n for any two vertices u and v with d(u,v)=2, then G is k-ordered hamiltonian. Our result implies the theorem of G. Chen et al. [Ars Combin. 70 (2004) 245-255] [1], which requires the degree sum condition for all pairs of non-adjacent vertices, not just those distance 2 apart.     

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 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)).     

Bandwidth on AT-free graphs

We study the classical Bandwidth problem from the viewpoint of parametrised algorithms. Given a graph G=(V,E) and a positive integer k, the Bandwidth problem asks whether there exists a bijective function β:{1,…,∣V∣}→V such that for every edge uv∈E, ∣β⁻¹(u)−β⁻¹(v)∣≤k. It is known that under standard complexity assumptions, no algorithm for Bandwidth with running time of the form f(k)n^O(1) exists, even when the input is restricted to trees. We initiate the search for classes of graphs where such algorithms do exist. We present an algorithm with running time n⋅2^O(klogk) for Bandwidth on AT-free graphs, a well-studied graph class that contains interval, permutation, and cocomparability graphs. Our result is the first non-trivial algorithm that shows fixed-parameter tractability of Bandwidth on a graph class on which the problem remains NP-complete.     

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.     

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.     

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 the profile of the corona of two graphs

The concept of profile, together with bandwidth, originates from handling sparse matrices in solving linear systems of equations. Given a graph G, the profile minimization problem is to find a one-to-one mapping f:V(G)→{1,2,…,|V(G)|} such that ∑_v∈V(G)max_x∈N[v](f(v)−f(x)) is as small as possible, where N[v]={v}∪{x:x is adjacent to v}. This paper studies the profile of the corona G∧H of two graphs G and H. In particular, bounds for the profile of the corona of two graphs are established. Also, exact values of the profiles of coronas G∧H are obtained when G has certain properties, including when G is a caterpillar, a complete graph or a cycle.     

Group path covering and distance two labeling of graphs

For a positive integer d, an L(d,1)-labeling f of a graph G is an assignment of integers to the vertices of G such that |f(u)−f(v)|?d if uv∈E(G), and |f(u)−f(v)|?1 if u and u are at distance two. The span of an L(d,1)-labeling f of a graph is the absolute difference between the maximum and minimum integers used by f. The L(d,1)-labeling number of G, denoted by λd,1(G), is the minimum span over all L(d,1)-labelings of G. An L^′(d,1)-labeling of a graph G is an L(d,1)-labeling of G which assigns different labels to different vertices. Denote by the L^′(d,1)-labeling number of G. Georges et al. [Discrete Math. 135 (1994) 103-111] established relationship between the L(2,1)-labeling number of a graph G and the path covering number of Gc, the complement of G. In this paper we first generalize the concept of the path covering of a graph to the t-group path covering. Then we establish the relationship between the L^′(d,1)-labeling number of a graph G and the (d−1)-group path covering number of Gc. Using this result, we prove that and for bipartite graphs G can be computed in polynomial time.     

On Clark and Suen bound-type results for k-domination and Roman domination of Cartesian product graphs

A set S of vertices of a graph G is a dominating set for G if every vertex of G is adjacent to at least one vertex of S. The domination number γ(G), of G, is the minimum cardinality of a dominating set in G. Moreover, if the maximum degree of G is Δ, then for every positive integer k≤Δ, the set S is a k-dominating set in G if every vertex outside of S is adjacent to at least k vertices of S. The k-domination number of G, denoted by γ _k (G), is the minimum cardinality of a k-dominating set in G. A map f: V→<texlscub>0, 1, 2</texlscub>is a Roman dominating function for G if for every vertex v with f(v)=0, there exists a vertex u∈N(v) such that f(u)=2. The weight of a Roman dominating function is f(V)=∑ _u∈V f(u). The Roman domination number γ_R(G), of G, is the minimum weight of a Roman dominating function on G. In this paper, we obtain that for any two graphs G and H, the k-domination number of the Cartesian product of G and H is bounded below by γ(G)γ _k (H)/2. Also, we obtain that the domination number of Cartesian product of G and H is bounded below by γ(G)γ_R(H)/3.     

Edge-bipancyclicity of the k-ary n-cubes with faulty nodes and edges

The k-ary n-cube has been one of the most popular interconnection networks for massively parallel systems. In this paper, we investigate the edge-bipancyclicity of k-ary n-cubes with faulty nodes and edges. It is proved that every healthy edge of the faulty k-ary n-cube with f_v faulty nodes and f_e faulty edges lies in a fault-free cycle of every even length from 4 to kⁿ − 2f_v (resp. kⁿ − f_v) if k ? 4 is even (resp. k ? 3 is odd) and f_v + f_e ? 2n − 3. The results are optimal with respect to the number of node and edge faults tolerated.     

A new approach for approximating node deletion problems

We present a new approach for approximating node deletion problems by combining the local ratio and the greedy multicovering algorithms. For a function , our approach allows to design a 2+max_v∈V(G)logf(v) approximation algorithm for the problem of deleting a minimum number of nodes so that the degree of each node v in the remaining graph is at most f(v). This approximation ratio is shown to be asymptotically optimal. The new method is also used to design a 1+(log2)(k−1) approximation algorithm for the problem of deleting a minimum number of nodes so that the remaining graph contains no k-bicliques.     

Weighted Coloring: further complexity and approximability results

Given a vertex-weighted graph G=(V,E;w), w(v)?0 for any v∈V, we consider a weighted version of the coloring problem which consists in finding a partition S=(S₁,…,Sk) of the vertex set of G into stable sets and minimizing where the weight of S is defined as . In this paper, we continue the investigation of the complexity and the approximability of this problem by answering some of the questions raised by Guan and Zhu [D.J. Guan, X. Zhu, A coloring problem for weighted graphs, Inform. Process. Lett. 61 (2) (1997) 77-81].     

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.     

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.     

Constructing the nearly shortest path in crossed cubes

A graph G is panconnected if each pair of distinct vertices u,v∈V(G) are joined by a path of length l for all d_G(u,v)?l?|V(G)|-1, where d_G(u,v) is the length of a shortest path joining u and v in G. Recently, Fan et. al. [J. Fan, X. Lin, X. Jia, Optimal path embedding in crossed cubes, IEEE Trans. Parall. Distrib. Syst. 16 (2) (2005) 1190-1200, J. Fan, X. Jia, X. Lin, Complete path embeddings in crossed cubes, Inform. Sci. 176 (22) (2006) 3332-3346] and Xu et. al. [J.M. Xu, M.J. Ma, M. Lu, Paths in Möbius cubes and crossed cubes, Inform. Proc. Lett. 97 (3) (2006) 94-97] both proved that n-dimensional crossed cube, CQ_n, is almost panconnected except the path of length d_CQn(u,v)+1 for any two distinct vertices u,v∈V(CQ_n). In this paper, we give a necessary and sufficient condition to check for the existence of paths of length d_CQn(u,v)+1, called the nearly shortest paths, for any two distinct vertices u,v in CQ_n. Moreover, we observe that only some pair of vertices have no nearly shortest path and we give a construction scheme for the nearly shortest path if it exists.     

Pancyclicity of ternary n-cube networks under the conditional fault model

A graph G is said to be conditional k-edge-fault pancyclic if after removing k faulty edges from G, under the assumption that each vertex is incident to at least two fault-free edges, the resulting graph contains a cycle of every length from its girth to |V(G)|. In this paper, we consider ternary n-cube networks and show that they are conditional (4n−5)-edge-fault pancyclic.