一类直径为2的极大平面图的Mostar指数

图G的Mostar指数定义为Mo(G)=∑uv∈Ε(G)|nu-nv|,其中nu表示在G中到顶点u的距离比到顶点v的距离近的顶点个数,nv表示到顶点v的距离比到顶点u的距离近的顶点个数.若一个图G的任两点之间的距离至多为2,且不是完全图,则称G是一个直径为2的图.已知直径为2点数至少为4的极大平面图的最小度为3或4.本文研究了直径为2且最小度为4的极大平面图的Mostar指数.具体说,若G是一个点数为n,直径为2,最小度为4的极大平面图,则(1)当n≤12时,Mostar指数被完全确定;(2)当n≥13时,4/3n2-44/3n+94/3≤Mo(G)≤2n2-16n+24,且达到上,下界的极图同时被找到.     

Clique partitions of distance multigraphs

We consider the minimum number of cliques needed to partition the edge set of D(G), the distance multigraph of a simple graph G. Equivalently, we seek to minimize the number of elements needed to label the vertices of a simple graph G by sets so that the distance between two vertices equals the cardinality of the intersection of their labels. We use a fractional analogue of this parameter to find lower bounds for the distance multigraphs of various classes of graphs. Some of the bounds are shown to be exact.     

A solitaire game played on 2-colored graphs

We introduce a solitaire game played on a graph. Initially one disk is placed at each vertex, one face green and the other red, oriented with either color facing up. Each move of the game consists of selecting a vertex whose disk shows green, flipping over the disks at neighboring vertices, and deleting the selected vertex. The game is won if all vertices are eliminated. We derive a simple parity-based necessary condition for winnability of a given game instance. By studying graph operations that construct new graphs from old ones, we obtain broad classes of graphs where this condition also suffices, thus characterizing the winnable games on such graphs. Concerning two familiar (but narrow) classes of graphs, we show that for trees a game is winnable if and only if the number of green vertices is odd, and for n-cubes a game is winnable if and only if the number of green vertices is even and not all vertices have the same color. We provide a linear-time algorithm for deciding winnability for games on maximal outerplanar graphs. We reduce the decision problem for winnability of a game on an arbitrary graph G to winnability of games on its blocks, and to winnability on homeomorphic images of G obtained by contracting edges at 2-valent vertices.     

The bondage number of graphs on topological surfaces and Teschner’s conjecture

The bondage number of a graph is the smallest number of its edges whose removal results in a graph having a larger domination number. We provide constant upper bounds for the bondage number of graphs on topological surfaces, and improve upper bounds for the bondage number in terms of the maximum vertex degree and the orientable and non-orientable genera of graphs. Also, we present stronger upper bounds for graphs with no triangles and graphs with the number of vertices larger than a certain threshold in terms of graph genera. This settles Teschner’s Conjecture in affirmative for almost all graphs. As an auxiliary result, we show tight lower bounds for the number of vertices of graphs 2-cell embeddable on topological surfaces of a given genus.     

The cover pebbling number of graphs

A pebbling move on a graph consists of taking two pebbles off of one vertex and placing one pebble on an adjacent vertex. In the traditional pebbling problem we try to reach a specified vertex of the graph by a sequence of pebbling moves. In this paper we investigate the case when every vertex of the graph must end up with at least one pebble after a series of pebbling moves. The cover pebbling number of a graph is the minimum number of pebbles such that however the pebbles are initially placed on the vertices of the graph we can eventually put a pebble on every vertex simultaneously. We find the cover pebbling numbers of trees and some other graphs. We also consider the more general problem where (possibly different) given numbers of pebbles are required for the vertices.     

Domination versus disjunctive domination in graphs

A dominating set in a graph G is a set S of vertices of G such that every vertex not in S is adjacent to a vertex of S. The domination number of G is the minimum cardinality of a dominating set of G. For a positive integer b, a set S of vertices in a graph G is a b-disjunctive dominating set in G if every vertex v not in S is adjacent to a vertex of S or has at least b vertices in S at distance 2 from it in G. The b-disjunctive domination number of G is the minimum cardinality of a b-disjunctive dominating set. In this paper, we continue the study of disjunctive domination in graphs. We present properties of b-disjunctive dominating sets in a graph. A characterization of minimal b-disjunctive dominating sets is given. We obtain bounds on the ratio of the domination number and the b-disjunctive domination number for various families of graphs, including regular graphs and trees.     

Dominator Colorings in Some Classes of Graphs

A dominator coloring is a coloring of the vertices of a graph such that every vertex is either alone in its color class or adjacent to all vertices of at least one other class. We present new bounds on the dominator coloring number of a graph, with applications to chordal graphs. We show how to compute the dominator coloring number in polynomial time for P ₄-free graphs, and we give a polynomial-time characterization of graphs with dominator coloring number at most 3.     

Looseness of Plane Graphs

A face of a vertex coloured plane graph is called loose if the number of colours used on its vertices is at least three. The looseness of a plane graph G is the minimum k such that any surjective k-colouring involves a loose face. In this paper we prove that the looseness of a connected plane graph G equals the maximum number of vertex disjoint cycles in the dual graph G* increased by 2. We also show upper bounds on the looseness of graphs based on the number of vertices, the edge connectivity, and the girth of the dual graphs. These bounds improve the result of Negami for the looseness of plane triangulations. We also present infinite classes of graphs where the equalities are attained.     

Independent sets and 2‐factors in edge‐chromatic‐critical graphs

In 1968, Vizing made the following two conjectures for graphs which are critical with respect to the chromatic index: (1) every critical graph has a 2‐factor, and (2) every independent vertex set in a critical graph contains at most half of the vertices. We prove both conjectures for critical graphs with many edges, and determine upper bounds for the size of independent vertex sets in those graphs. © 2003 Wiley Periodicals, Inc. J Graph Theory 45: 113–118, 2004     

A Characterization of Distance-Regular Graphs with Diameter Three

We characterize the distance-regular graphs with diameter three by giving an expression for the number of vertices at distance two from each given vertex, in terms of the spectrum of the graph.     

An adjacency lemma for critical multigraphs

In edge colouring it is often useful to have information about the degree distribution of the neighbours of a given vertex. For example, the well-known Vizing's Adjacency Lemma provides a useful lower bound on the number of vertices of maximum degree adjacent to a given one in a critical graph. We consider an extension of this problem, where we seek information on vertices at distance two from a given vertex in a critical graph. We extend and, simultaneously, generalize to multigraphs two results proved, respectively, by Zhang [Every planar graph with maximum degree seven is Class 1, Graphs Combin. 16 (2000) 467-495] and Sanders and Zhao [Planar graphs of maximum degree seven are Class 1, J. Combin. Theory Ser. B 83 (2001) 201-212].     

On Narrow Hexagonal Graphs with a 3-Homogeneous Suborbit

A connected graph of girth m 3 is called a polygonal graph if it contains a set of m-gons such that every path of length two is contained in a unique element of the set. In this paper we investigate polygonal graphs of girth 6 or more having automorphism groups which are transitive on the vertices and such that the vertex stabilizers are 3-homogeneous on adjacent vertices. We previously showed that the study of such graphs divides naturally into a number of substantial subcases. Here we analyze one of these cases and characterize the k-valent polygonal graphs of girth 6 which have automorphism groups transitive on vertices, which preserve the set of special hexagons, and which have a suborbit of size k – 1 at distance three from a given vertex.     

Coloring, location and domination of corona graphs

A vertex coloring of a graph G is an assignment of colors to the vertices of G so that every two adjacent vertices of G have different colors. A coloring related property of a graphs is also an assignment of colors or labels to the vertices of a graph, in which the process of labeling is done according to an extra condition. A set S of vertices of a graph G is a dominating set in G if every vertex outside of S is adjacent to at least one vertex belonging to S. A domination parameter of G is related to those structures of a graph that satisfy some domination property together with other conditions on the vertices of G. In this article we study several mathematical properties related to coloring, domination and location of corona graphs. We investigate the distance-k colorings of corona graphs. Particularly, we obtain tight bounds for the distance-2 chromatic number and distance-3 chromatic number of corona graphs, through some relationships between the distance-k chromatic number of corona graphs and the distance-k chromatic number of its factors. Moreover, we give the exact value of the distance-k chromatic number of the corona of a path and an arbitrary graph. On the other hand, we obtain bounds for the Roman dominating number and the locating–domination number of corona graphs. We give closed formulaes for the k-domination number, the distance-k domination number, the independence domination number, the domatic number and the idomatic number of corona graphs.     

Enumerating maximal independent sets with applications to graph colouring

We give tight upper bounds on the number of maximal independent sets of size k (and at least k and at most k) in graphs with n vertices. As an application of the proof, we construct improved algorithms for graph colouring and computing the chromatic number of a graph.     

Upper bounds on vertex distinguishing chromatic index of some Halin graphs

A vertex distinguishing edge coloring of a graph G is a proper edge coloring of G such that any pair of vertices has the distinct sets of colors. The minimum number of colors required for a vertex distinguishing edge coloring of a graph G is denoted by ???? _s (G). In this paper, we obtained upper bounds on the vertex distinguishing chromatic index of 3-regular Halin graphs and Halin graphs with ??(G) ?? 4, respectively.     

Randomized greedy algorithms for finding small k‐dominating sets of regular graphs

A k‐dominating set of a graph G is a subset ?? of the vertices of G such that every vertex of G is either in ?? or at distance at most k from a vertex in ??. It is of interest to find k‐dominating sets of small cardinality. In this paper we consider simple randomized greedy algorithms for finding small k‐dominating sets of regular graphs. We analyze the average‐case performance of the most efficient of these simple heuristics showing that it performs surprisingly well on average. The analysis is performed on random regular graphs using differential equations. This, in turn, proves upper bounds on the size of a minimum k‐dominating set of random regular graphs. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 22, 2005     

Bounds on the Rubbling and Optimal Rubbling Numbers of Graphs

A pebbling move on a graph removes two pebbles at a vertex and adds one pebble at an adjacent vertex. Rubbling is a version of pebbling where an additional move is allowed. In this new move, one pebble each is removed at vertices v and w adjacent to a vertex u, and an extra pebble is added at vertex u. A vertex is reachable from a pebble distribution if it is possible to move a pebble to that vertex using rubbling moves. The rubbling number is the smallest number m needed to guarantee that any vertex is reachable from any pebble distribution of m pebbles. The optimal rubbling number is the smallest number m needed to guarantee a pebble distribution of m pebbles from which any vertex is reachable. We give bounds for rubbling and optimal rubbling numbers. In particular, we find an upper bound for the rubbling number of n-vertex, diameter d graphs, and estimates for the maximum rubbling number of diameter 2 graphs. We also give a sharp upper bound for the optimal rubbling number, and sharp upper and lower bounds in terms of the diameter.     

Partitioning the Vertices of a Graph into Two Total Dominating Sets

A total dominating set in a graph G is a set S of vertices of G such that every vertex in G is adjacent to a vertex of S. We study graphs whose vertex set can be partitioned into two total dominating sets. In particular, we develop several sufficient conditions for a graph to have a vertex partition into two total dominating sets. We also show that with the exception of the cycle on five vertices, every selfcomplementary graph with minimum degree at least two has such a partition.     

The disjunctive domination number of a graph

Abstract

For a positive integer b, we define a set S of vertices in a graph G as a b-disjunctive dominating set if every vertex not in S is adjacent to a vertex of S or has at least b vertices in S at distance 2 from it. The b-disjunctive domination number is the minimum cardinality of such a set. This concept is motivated by the concepts of distance domination and exponential domination. In this paper, we start with some simple results, then establish bounds on the parameter especially for regular graphs and claw-free graphs. We also show that determining the parameter is NP-complete, and provide a linear-time algorithm for trees.     

Class Steiner trees and VLSI-design

We consider a generalized version of the Steiner problem in graphs, motivated by the wire routing phase in physical VLSI design: given a connected, undirected distance graph with required classes of vertices and Steiner vertices, find a shortest connected subgraph containing at least one vertex of each required class. We show that this problem is NP-hard, even if there are no Steiner vertices and the graph is a tree. Moreover, the same complexity result holds if the input class Steiner graph additionally is embedded in a unit grid, if each vertex has degree at most three, and each class consists of no more than three vertices. For similar restricted versions, we prove MAX SNP-hardness and we show that there exists no polynomial-time approximation algorithm with a constant bound on the relative error, unless P = NP. We propose two efficient heuristics computing different approximate solutions in time O(¦E¦+¦V¦log¦V¦) and in time O(c(¦E¦+¦V¦log¦V¦)), respectively, where E is the set of edges in the given graph, V is the set of vertices, and c is the number of classes. We present some promising implementation results. kw]Steiner Tree; Heuristic; Approximation complexity; MAX-SNP-hardness