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.     

Total restrained domination in graphs

In this paper, we initiate the study of a variation of standard domination, namely total restrained domination. Let G=(V,E) be a graph. A set D⊆V is a total restrained dominating set if every vertex in V−D has at least one neighbor in D and at least one neighbor in V−D, and every vertex in D has at least one neighbor in D. The total restrained domination number of G, denoted by γ_tr(G), is the minimum cardinality of all total restrained dominating sets of G. We determine the best possible upper and lower bounds for γ_tr(G), characterize those graphs achieving these bounds and find the best possible lower bounds for where both G and are connected.     

Conditional fault hamiltonian connectivity of the complete graph

A path in G is a hamiltonian path if it contains all vertices of G. A graph G is hamiltonian connected if there exists a hamiltonian path between any two distinct vertices of G. The degree of a vertex u in G is the number of vertices of G adjacent to u. We denote by δ(G) the minimum degree of vertices of G. A graph G is conditional k edge-fault tolerant hamiltonian connected if G−F is hamiltonian connected for every F⊂E(G) with |F|?k and δ(G−F)?3. The conditional edge-fault tolerant hamiltonian connectivity is defined as the maximum integer k such that G is k edge-fault tolerant conditional hamiltonian connected if G is hamiltonian connected and is undefined otherwise. Let n?4. We use Kn to denote the complete graph with n vertices. In this paper, we show that for n∉{4,5,8,10}, , , , and .     

Enhanced algorithms for Local Search

Let G=(V,E) be a finite graph, and be any function. The Local Search problem consists in finding a local minimum of the function f on G, that is a vertex v such that f(v) is not larger than the value of f on the neighbors of v in G. In this note, we first prove a separation theorem slightly stronger than the one of Gilbert, Hutchinson and Tarjan for graphs of constant genus. This result allows us to enhance a previously known deterministic algorithm for Local Search with query complexity , so that we obtain a deterministic query complexity of , where n is the size of G, d is its maximum degree, and g is its genus. We also give a quantum version of our algorithm, whose query complexity is of . Our deterministic and quantum algorithms have query complexities respectively smaller than the algorithm Randomized Steepest Descent of Aldous and Quantum Steepest Descent of Aaronson for large classes of graphs, including graphs of bounded genus and planar graphs.     

Note on the connectivity of line graphs

Let G be a connected graph with vertex set V(G), edge set E(G), vertex-connectivity κ(G) and edge-connectivity λ(G).A subset S of E(G) is called a restricted edge-cut if G−S is disconnected and each component contains at least two vertices. The restricted edge-connectivity λ₂(G) is the minimum cardinality over all restricted edge-cuts. Clearly λ₂(G)?λ(G)?κ(G).In 1969, Chartrand and Stewart have shown that

     

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.     

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.     

On domination number of Cartesian product of directed cycles

Let γ(G) denote the domination number of a digraph G and let Cm□Cn denote the Cartesian product of Cm and Cn, the directed cycles of length m,n?2. In this paper, we determine the exact values: γ(C₂□Cn)=n; γ(C₃□Cn)=n if , otherwise, γ(C₃□Cn)=n+1; if , otherwise, .     

2-local 5/4-competitive algorithm for multicoloring triangle-free hexagonal graphs

An important optimization problem in the design of cellular networks is to assign sets of frequencies to transmitters to avoid unacceptable interference. A cellular network is generally modeled as a subgraph of the infinite triangular lattice. The distributed frequency assignment problem can be abstracted as a multicoloring problem on a weighted hexagonal graph, where the weight vector represents the number of calls to be assigned at vertices. In this paper we present a 2-local distributed algorithm for multicoloring triangle-free hexagonal graphs using only the local clique numbers ω₁(v) and ω₂(v) at each vertex v of the given hexagonal graph, which can be computed from local information available at the vertex. We prove that the algorithm uses no more than colors for any triangle-free hexagonal graph G, without explicitly computing the global clique number ω(G). Hence the competitive ratio of the algorithm is 5/4.     

The Pair Completion algorithm for the Homogeneous Set Sandwich Problem

A homogeneous set is a non-trivial module of a graph, i.e., a non-empty, non-unitary, proper vertex subset such that all its elements present the same outer neighborhood. Given two graphs G₁(V,E₁) and G₂(V,E₂), the Homogeneous Set Sandwich Problem (HSSP) asks whether there exists a graph GS(V,ES), E₁⊆ES⊆E₂, which has a homogeneous set. This paper presents an algorithm that uses the concept of bias graph [S. Tang, F. Yeh, Y. Wang, An efficient algorithm for solving the homogeneous set sandwich problem, Inform. Process. Lett. 77 (2001) 17-22] to solve the problem in time, thus outperforming the other known HSSP deterministic algorithms for inputs where .     

Adjacent vertex-distinguishing edge and total chromatic numbers of hypercubes

An adjacent vertex-distinguishing edge coloring of a simple graph G is a proper edge coloring of G such that incident edge sets of any two adjacent vertices are assigned different sets of colors. A total coloring of a graph G is a coloring of both the edges and the vertices. A total coloring is proper if no two adjacent or incident elements receive the same color. An adjacent vertex-distinguishing total coloring h of a simple graph G=(V,E) is a proper total coloring of G such that H(u)≠H(v) for any two adjacent vertices u and v, where H(u)={h(wu)|wu∈E(G)}∪{h(u)} and H(v)={h(xv)|xv∈E(G)}∪{h(v)}. The minimum number of colors required for an adjacent vertex-distinguishing edge coloring (resp. an adjacent vertex-distinguishing total coloring) of G is called the adjacent vertex-distinguishing edge chromatic number (resp. adjacent vertex-distinguishing total chromatic number) of G and denoted by (resp. χat(G)). In this paper, we consider the adjacent vertex-distinguishing edge chromatic number and adjacent vertex-distinguishing total chromatic number of the hypercube Qn, prove that for n?3 and χat(Qn)=Δ(Qn)+2 for n?2.     

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 edge covering colorings of graphs

An edge covering coloring of a graph G is an edge-coloring of G such that each color appears at each vertex at least one time. The maximum integer k such that G has an edge covering coloring with k colors is called the edge covering chromatic index of G and denoted by . It is known that for any graph G with minimum degree δ(G), it holds that . Based on the subgraph of G induced by the vertices of minimum degree, we find a new sufficient condition for a graph G to satisfy . This result substantially extends a result of Wang et al. in 2006.