On anti-magic labeling for graph products

An anti-magic labeling of a finite simple undirected graph with p vertices and q edges is a bijection from the set of edges to the set of integers {1,2,…,q} such that the vertex sums are pairwise distinct, where the vertex sum at one vertex is the sum of labels of all edges incident to such vertex. A graph is called anti-magic if it admits an anti-magic labeling. Hartsfield and Ringel conjectured in 1990 that all connected graphs except K₂ are anti-magic. Recently, Alon et al. showed that this conjecture is true for dense graphs, i.e. it is true for p-vertex graphs with minimum degree Ω(logp). In this article, new classes of sparse anti-magic graphs are constructed through Cartesian products and lexicographic products.     

An upper bound on the size of a largest clique in a graph

A graph is point determining if distinct vertices have distinct neighborhoods. The nucleus of a point-determining graph is the set G^O of all vertices, v, such that G–v is point determining. In this paper we show that the size, ω(G), of a maximum clique in G satisfies ω(G) ? 2|π (G)^O|, where π(G) (the point determinant of G) is obtained from G by identifying vertices which have the same neighborhood.     

On the Neighbour-Distinguishing Index of a Graph

A proper edge colouring of a graph G is neighbour-distinguishing provided that it distinguishes adjacent vertices by sets of colours of their incident edges. It is proved that for any planar bipartite graph G with Δ(G)≥12 there is a neighbour-distinguishing edge colouring of G using at most Δ(G)+1 colours. Colourings distinguishing pairs of vertices that satisfy other requirements are also considered.     

Proof of a conjecture on game domination

Let G(V, E) be a simple, undirected graph where V is the set of vertices and E is the set of edges. A b‐dimensional cube is a Cartesian product I₁×I₂×···×I_b, where each I_i is a closed interval of unit length on the real line. The cubicity of G, denoted by cub(G), is the minimum positive integer b such that the vertices in G can be mapped to axis parallel b‐dimensional cubes in such a way that two vertices are adjacent in G if and only if their assigned cubes intersect. An interval graph is a graph that can be represented as the intersection of intervals on the real line—i.e. the vertices of an interval graph can be mapped to intervals on the real line such that two vertices are adjacent if and only if their corresponding intervals overlap. Suppose S(m) denotes a star graph on m+1 nodes. We define claw number ψ(G) of the graph to be the largest positive integer m such that S(m) is an induced subgraph of G. It can be easily shown that the cubicity of any graph is at least ?log₂ψ(G)?. In this article, we show that for an interval graph G ?log₂ψ(G)??cub(G)??log₂ψ(G)?+2. It is not clear whether the upper bound of ?log₂ψ(G)?+2 is tight: till now we are unable to find any interval graph with cub(G)>?log₂ψ(G)?. We also show that for an interval graph G, cub(G)??log₂α?, where α is the independence number of G. Therefore, in the special case of ψ(G)=α, cub(G) is exactly ?log₂α₂?. The concept of cubicity can be generalized by considering boxes instead of cubes. A b‐dimensional box is a Cartesian product I₁×I₂×···×I_b, where each I_i is a closed interval on the real line. The boxicity of a graph, denoted box(G), is the minimum k such that G is the intersection graph of k‐dimensional boxes. It is clear that box(G)?cub(G). From the above result, it follows that for any graph G, cub(G)?box(G)?log₂α?. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 323–333, 2010     

Graphs with constant link and small degree or order

The graph G has constant link L if for each vertex x of G the graph induced by G on the vertices adjacent to x is isomorphic to L. For each graph L on 6 or fewer vertices we decide whether or not there exists a graph G with constant link L. From this we are able to list all graphs on 11 or fewer vertices which have constant link.     

Exotic n-universal graphs

An n-universal graph is a graph that contains as an induced subgraph a copy of every graph on n vertices. It is shown that for each positive integer n > 1 there exists an n-universal graph G on 4ⁿ - 1 vertices such that G is a (v, k, λ)-graph, and both G and its complement G¯ are 1-transitive in the sense of W. T. Tutte and are of diameter 2. The automorphism group of G is a transitive rank 3 permutation group, i.e., it acts transitively on (1) the vertices of G, (2) the ordered pairs uv of adjacent vertices of G, and (3) the ordered pairs xy of nonadjacent vertices of G.     

A Structure Theorem for Strong Immersions

A graph H is strongly immersed in G if H is obtained from G by a sequence of vertex splittings (i.e., lifting some pairs of incident edges and removing the vertex) and edge removals. Equivalently, vertices of H are mapped to distinct vertices of G (branch vertices) and edges of H are mapped to pairwise edge‐disjoint paths in G, each of them joining the branch vertices corresponding to the ends of the edge and not containing any other branch vertices. We describe the structure of graphs avoiding a fixed graph as a strong immersion. The theorem roughly states that a graph which excludes a fixed graph as a strong immersion has a tree‐like decomposition into pieces glued together on small edge cuts such that each piece of the decomposition has a path‐like linear decomposition isolating the high degree vertices.     

On the Convexity Number of Graphs

A set of vertices S in a graph is convex if it contains all vertices which belong to shortest paths between vertices in S. The convexity number c(G) of a graph G is the maximum cardinality of a convex set of vertices which does not contain all vertices of G. We prove NP-completeness of the problem to decide for a given bipartite graph G and an integer k whether c(G) ≥ k. Furthermore, we identify natural necessary extension properties of graphs of small convexity number and study the interplay between these properties and upper bounds on the convexity number.     

Cyclability of 3‐connected graphs

A subset S of vertices of a graph G is called cyclable in G if there is in G some cycle containing all the vertices of S. We denote by α(S, G) the number of vertices of a maximum independent set of G[S]. We prove that if G is a 3‐connected graph or order n and if S is a subset of vertices such that the degree sum of any four independent vertices of S is at least n + 2α(S, G) −2, then S is cyclable. This result implies several known results on cyclability or Hamiltonicity. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 191–203, 2000     

Induced Decompositions of Graphs

We consider those graphs G that admit decompositions into copies of a fixed graph F, each copy being an induced subgraph of G. We are interested in finding the extremal graphs with this property, that is, those graphs G on n vertices with the maximum possible number of edges. We discuss the cases where F is a complete equipartite graph, a cycle, a star, or a graph on at most four vertices.     

Almost self-centered graphs

The center of a graph is the set of vertices with minimum eccentricity. Graphs in which all vertices are central are called self-centered graphs. In this paper almost self-centered (ASC) graphs are introduced as the graphs with exactly two non-central vertices. The block structure of these graphs is described and constructions for generating such graphs are proposed. Embeddings of arbitrary graphs into ASC graphs are studied. In particular it is shown that any graph can be embedded into an ASC graph of prescribed radius. Embeddings into ASC graphs of radius two are studied in more detail. ASC index of a graph G is introduced as the smallest number of vertices needed to add to G such that G is an induced subgraph of an ASC graph.     

The geodetic number of the lexicographic product of graphs

A set S of vertices of a graph G is a geodetic set if every vertex of G lies in an interval between two vertices from S. The size of a minimum geodetic set in G is the geodetic number g(G) of G. We find that the geodetic number of the lexicographic product G°H for a non-complete graph H lies between 2 and 3g(G). We characterize the graphs G and H for which g(G°H)=2, as well as the lexicographic products T°H that enjoy g(T°H)=3g(G), when T is isomorphic to a tree. Using a new concept of the so-called geodominating triple of a graph G, a formula that expresses the exact geodetic number of G°H is established, where G is an arbitrary graph and H a non-complete graph.     

Common Divisor Character Degree Graphs of Solvable Groups with Four Vertices

Let G be a finite solvable group. The common divisor graph Γ(G) attached to G is a character degree graph. Its vertices are the degrees of the nonlinear irreducible complex characters of G, and different vertices m, n are adjacent if the greatest common divisor (m, n) > 1. In this article, we classify all graphs with four vertices that may occur as Γ(G) for solvable group G.     

New Bounds on the Grundy Number of Products of Graphs

The Grundy number of a graph G is the largest k such that G has a greedy k‐coloring, that is, a coloring with k colors obtained by applying the greedy algorithm according to some ordering of the vertices of G. In this article, we give new bounds on the Grundy number of the product of two graphs. © 2011 Wiley Periodicals, Inc. J Graph Theory 71:78–88, 2012     

Grid Embedding of 4-Connected Plane Graphs

A straight line grid embedding of a plane graph G is a drawing of G such that the vertices are drawn at grid points and the edges are drawn as nonintersecting straight line segments. In this paper we show that if a 4-connected plane graph G has at least four vertices on its external face, then G can be embedded on a grid of size such that and , where n is the number of vertices of G. Such an embedding can be computed in linear time. Received March 30, 1995, and in revised form January 3, 1996.     

Edge-clique graphs

The edge-clique graphK(G) of a graphG is that graph whose vertices correspond to the edges ofG and where two vertices ofK(G) are adjacent whenever the corresponding edges ofG belong to a common clique. It is shown that every edge-clique graph is a clique graph, and that ifG is either an interval graph or a line graph, then so too isK(G). An algorithm is provided for determining whether a graph is an edge-clique graph. A new graph called the STP graph is introduced and a relationship involving this graph, the edge-clique graph, and the line graph is presented. The STP graphs are also characterized.Research supported in part by Office of Naval Research Contract N00014-88-K-0018.Research supported in part by Office of Naval Research Contract N00014-88-K-0163.     

Closure Concepts: A Survey

In this paper we survey results of the following type (known as closure results). Let P be a graph property, and let C(u,v) be a condition on two nonadjacent vertices u and v of a graph G. Then G+uv has property P if and only if G has property P. The first and now well-known result of this type was established by Bondy and Chvátal in a paper published in 1976: If u and v are two nonadjacent vertices with degree sum n in a graph G on n vertices, then G+uv is hamiltonian if and only if G is hamiltonian. Based on this result, they defined the n-closure cl_n (G) of a graph G on n vertices as the graph obtained from G by recursively joining pairs of nonadjacent vertices with degree sum n until no such pair remains. They showed that cl_n(G) is well-defined, and that G is hamiltonian if and only if cl_n(G) is hamiltonian. Moreover, they showed that cl_n(G) can be obtained by a polynomial algorithm, and that a Hamilton cycle in cl_n(G) can be transformed into a Hamilton cycle of G by a polynomial algorithm. As a consequence, for any graph G with cl_n(G)=K _n (and n≥3), a Hamilton cycle can be found in polynomial time, whereas this problem is NP-hard for general graphs. All classic sufficient degree conditions for hamiltonicity imply a complete n-closure, so the closure result yields a common generalization as well as an easy proof for these conditions. In their first paper on closures, Bondy and Chvátal gave similar closure results based on degree sum conditions for nonadjacent vertices for other graph properties. Inspired by their first results, many authors developed other closure concepts for a variety of graph properties, or used closure techniques as a tool for obtaining deeper sufficiency results with respect to these properties. Our aim is to survey this progress on closures made in the past (more than) twenty years. Revised: September 27, 1999     

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.     

Fragile graphs with small independent cuts

A graph is called fragile if it has a vertex cut which is also an independent set. Chen and Yu proved that every graph with n vertices and at most 2n?4 edges is fragile, which was conjectured to be true by Caro. However, their proof does not give any information on the number of vertices in the independent cuts. The purpose of this paper is to investigate when a graph has a small independent cut. We show that if G is a graph on n vertices and at most (12n/7)?3 edges, then G contains an independent cut S with ∣S∣≤3. Upper bounds on the number of edges of a graph having an independent cut of size 1 or 2 are also obtained. We also show that for any positive integer k, there is a positive number ε such that there are infinitely many graphs G with n vertices and at most (2?ε)n edges, but G has no independent cut with less than k vertices. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 327–341, 2002     

A Note on Degree Conditions for Traceability in Locally Claw-Free Graphs

We say that a set S of vertices is traceable in a graph G whenever there is a path in G containing all vertices of S. In this paper we study the problem of traceability of a prescribed set of vertices in a locally claw-free graph (i.e. a graph in which some specified vertices are not centers of an induced claw). In particular we give sufficient degree conditions restricted to the given set S of vertices for the traceability of S.