The complexity of determining the rainbow vertex-connection of a graph

A vertex-colored graph is rainbow vertex-connected if any two vertices are connected by a path whose internal vertices have distinct colors, which was introduced by Krivelevich and Yuster. The rainbow vertex-connection of a connected graph G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertex-connected. In this paper, we study the complexity of determining the rainbow vertex-connection of a graph and prove that computing rvc(G) is NP-Hard. Moreover, we show that it is already NP-Complete to decide whether rvc(G)=2. We also prove that the following problem is NP-Complete: given a vertex-colored graph G, check whether the given coloring makes G rainbow vertex-connected.     

A new sufficient condition for Hamiltonicity of graphs

Rahman and Kaykobad proved the following theorem on Hamiltonian paths in graphs. Let G be a connected graph with n vertices. If d(u)+d(v)+δ(u,v)?n+1 for each pair of distinct non-adjacent vertices u and v in G, where δ(u,v) is the length of a shortest path between u and v in G, then G has a Hamiltonian path. It is shown that except for two families of graphs a graph is Hamiltonian if it satisfies the condition in Rahman and Kaykobad's theorem. The result obtained in this note is also an answer for a question posed by Rahman and Kaykobad.     

A polynomial time algorithm for obtaining minimum edge ranking on two-connected outerplanar graphs

An edge ranking of a graph G is a labeling r of its edges with positive integers such that every path between two different edges eu, ev with the same rank r(eu)=r(ev) contains an intermediate edge ew with rank r(ew)>r(eu). An edge ranking of G is minimum if the largest rank k assigned is the smallest among all rankings of G. The edge ranking problem is to find a minimum edge ranking of given graph G. This problem is NP-hard and no polynomial time algorithm for solving it is known for non-trivial classes of graphs other than the class of trees. In this paper, we first show, on a general graph G, a relation between a minimum edge ranking of G and its minimal cuts, which ensures that we can obtain a polynomial time algorithm for obtaining minimum edge ranking of a given graph G if minimal cuts for each subgraph of G can be found in polynomial time and the number of those is polynomial. Based on this relation, we develop a polynomial time algorithm for finding a minimum edge ranking on a 2-connected outerplanar graph.     

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.     

On the oriented chromatic number of grids

In this paper, we focus on the oriented coloring of graphs. Oriented coloring is a coloring of the vertices of an oriented graph G without symmetric arcs such that (i) no two neighbors in G are assigned the same color, and (ii) if two vertices u and v such that (u,v)∈A(G) are assigned colors c(u) and c(v), then for any (z,t)∈A(G), we cannot have simultaneously c(z)=c(v) and c(t)=c(u). The oriented chromatic number of an unoriented graph G is the smallest number k of colors for which any of the orientations of G can be colored with k colors.The main results we obtain in this paper are bounds on the oriented chromatic number of particular families of planar graphs, namely 2-dimensional grids, fat trees and fat fat trees.     

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.     

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.     

An optimal parallel algorithm for c-vertex-ranking of trees

For a positive integer c, a c-vertex-ranking of a graph G=(V,E) is a labeling of the vertices of G with integers such that, for any label i, deletion of all vertices with labels >i leaves connected components, each having at most c vertices with label i. The c-vertex-ranking problem is to find a c-vertex-ranking of a given graph using the minimum number of ranks. In this paper we give an optimal parallel algorithm for solving the c-vertex-ranking problem on trees in O(log₂n) time using linear number of operations on the EREW PRAM model.     

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.     

The hub number of a graph

A hub set in a graph G is a set U⊆V(G) such that any two vertices outside U are connected by a path whose internal vertices lie in U. We prove that h(G)?hc(G)?γc(G)?h(G)+1, where h(G), hc(G), and γc(G), respectively, are the minimum sizes of a hub set in G, a hub set inducing a connected subgraph, and a connected dominating set. Furthermore, all graphs with γc(G)>hc(G)?4 are obtained by substituting graphs into three consecutive vertices of a cycle; this yields a polynomial-time algorithm to check whether hc(G)=γc(G).     

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.     

Consensus models: Computational complexity aspects in modern approaches to the list coloring problem

In the paper we study new approaches to the problem of list coloring of graphs. In the problem we are given a simple graph G=(V,E) and, for every v∈V, a nonempty set of integers S(v); we ask if there is a coloring c of G such that c(v)∈S(v) for every v∈V. Modern approaches, connected with applications, change the question—we now ask if S can be changed, using only some elementary transformations, to ensure that there is such a coloring and, if the answer is yes, what is the minimal number of changes. In the paper for studying the adding, the trading and the exchange models of list coloring, we use the following transformations:

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adding of colors (the adding model): select two vertices u, v and a color c∈S(u); add c to S(v), i.e. set S(v):=S(v)∪{c};

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trading of colors (the trading model): select two vertices u, v and a color c∈S(u); move c from S(u) to S(v), i.e. set S(u):=S(u)?{c} and S(v):=S(v)∪{c};

•

exchange of colors (the exchange model): select two vertices u, v and two colors c∈S(u), d∈S(v); exchange c with d, i.e. set S(u):=(S(u)?{c})∪{d} and S(v):=(S(v)?{d})∪{c}.

Our study focuses on computational complexity of the above models and their edge versions. We consider these problems on complete graphs, graphs with bounded cyclicity and partial k-trees, receiving in all cases polynomial algorithms or proofs of NP-hardness.     

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.     

The Hub Number of Sierpiński-Like Graphs

A set Q⊆V is a hub set of a graph G=(V,E) if, for every pair of vertices u,v∈V∖Q, there exists a path from u to v such that all intermediate vertices are in Q. The hub number of G is the minimum size of a hub set in G. This paper derives the hub numbers of Sierpiński-like graphs including: Sierpiński graphs, extended Sierpiński graphs, and Sierpiński gasket graphs. Meanwhile, the corresponding minimum hub sets are also obtained.     

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.     

Geodesic pancyclicity and balanced pancyclicity of Augmented cubes

For two distinct vertices u,v∈V(G), a cycle is called geodesic cycle with u and v if a shortest path of G joining u and v lies on the cycle; and a cycle C is called balanced cycle with u and v if dC(u,v)=max{dC(x,y)|x,y∈V(C)}. A graph G is pancyclic [J. Mitchem, E. Schmeichel, Pancyclic and bipancyclic graphs a survey, Graphs and applications (1982) 271-278] if it contains a cycle of every length from 3 to |V(G)| inclusive. A graph G is called geodesic pancyclic [H.C. Chan, J.M. Chang, Y.L. Wang, S.J. Horng, Geodesic-pancyclic graphs, in: Proceedings of the 23rd Workshop on Combinatorial Mathematics and Computation Theory, 2006, pp. 181-187] (respectively, balanced pancyclic) if for each pair of vertices u,v∈V(G), it contains a geodesic cycle (respectively, balanced cycle) of every integer length of l satisfying max{2dG(u,v),3}?l?|V(G)|. Lai et al. [P.L. Lai, J.W. Hsue, J.J.M. Tan, L.H. Hsu, On the panconnected properties of the Augmented cubes, in: Proceedings of the 2004 International Computer Symposium, 2004, pp. 1249-1251] proved that the n-dimensional Augmented cube, AQn, is pancyclic in the sense that a cycle of length l exists, 3?l?|V(AQn)|. In this paper, we study two new pancyclic properties and show that AQn is geodesic pancyclic and balanced pancyclic for n?2.     

On the Rainbow Connectivity of Graphs: Complexity and FPT Algorithms

For a graph G=(V,E) and a color set C, let f:E→C be an edge-coloring of G in which two adjacent edges may have the same color. Then, the graph G edge-colored by f is rainbow connected if every two vertices of G have a path in which all edges are assigned distinct colors. Chakraborty et al. defined the problem of determining whether the graph colored by a given edge-coloring is rainbow connected. Chen et al. introduced the vertex-coloring version of the problem as a variant, and we introduce the total-coloring version in this paper. We settle the precise computational complexities of all the three problems with regards to graph diameters, and also characterize these with regards to certain graph classes: cacti, outer planer and series-parallel graphs. We then give FPT algorithms for the three problems on general graphs when parameterized by the number of colors in C; our FPT algorithms imply that all the three problems can be solved in polynomial time for any graph with n vertices if |C|=O(logn).     

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.     

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.