Average distances and distance domination numbers

Let k be a positive integer and G be a simple connected graph with order n. The average distance μ(G) of G is defined to be the average value of distances over all pairs of vertices of G. A subset D of vertices in G is said to be a k-dominating set of G if every vertex of V(G)−D is within distance k from some vertex of D. The minimum cardinality among all k-dominating sets of G is called the k-domination number γ_k(G) of G. In this paper tight upper bounds are established for μ(G), as functions of n, k and γ_k(G), which generalizes the earlier results of Dankelmann [P. Dankelmann, Average distance and domination number, Discrete Appl. Math. 80 (1997) 21-35] for k=1.     

On an extension of distance hereditary graphs

Given a simple and finite connected graph G, the distance d_G(u,v) is the length of the shortest induced {u,v}-path linking the vertices u and v in G. Bandelt and Mulder [H.J. Bandelt, H.M. Mulder, Distance hereditary graphs, J. Combin. Theory Ser. B 41 (1986) 182-208] have characterized the class of distance hereditary graphs where the distance is preserved in each connected induced subgraph. In this paper, we are interested in the class of k-distance hereditary graphs (k∈N) which consists in a parametric extension of the distance heredity notion. We allow the distance in each connected induced subgraph to increase by at most k. We provide a characterization of k-distance hereditary graphs in terms of forbidden configurations for each k≥2.     

Some graft transformations and its applications on the distance spectral radius of a graph

Let D(G)=(d_i,j)_n×n denote the distance matrix of a connected graph G with order n, where d_ij is equal to the distance between v_i and v_j in G. The largest eigenvalue of D(G) is called the distance spectral radius of graph G, denoted by ?(G). In this paper, some graft transformations that decrease or increase ?(G) are given. With them, for the graphs with both order n and k pendant vertices, the extremal graphs with the minimum distance spectral radius are completely characterized; the extremal graph with the maximum distance spectral radius is shown to be a dumbbell graph (obtained by attaching some pendant edges to each pendant vertex of a path respectively) when 2≤k≤n−2; for k=1,2,3,n−1, the extremal graphs with the maximum distance spectral radius are completely characterized.     

Bandwidth of the corona of two graphs

The bandwidth B(G) of a graph G is the minimum of the quantity max{|f(u)-f(v)|:uv∈E(G)} taken over all injective integer numberings f of G. The corona of two graphs G and H, written as G°H, is the graph obtained by taking one copy of G and |V(G)| copies of H, and then joining the ith vertex of G to every vertex in the ith copy of H. In this paper, we investigate the bandwidth of the corona of two graphs. For a graph G, we denote the connectivity of G by κ(G). Let G be a graph on n vertices with B(G)=κ(G)=k?2 and let H be a graph of order m. Let c,p and q be three integers satisfying 1?c?k-1 and . We define h_i=(2k-1)m+(k-i)(⌊(2k-1)m/i⌋+1)+1 for i=1,2,…,k and b=max{⌈(n(m+1)-qm-1)/(p+2)⌉,⌈(n(m+1)+k-q-1)/(p+3)⌉}. Then, among other results, we prove that

     

The edge-Wiener index of a graph

If G is a connected graph, then the distance between two edges is, by definition, the distance between the corresponding vertices of the line graph of G. The edge-Wiener index W_e of G is then equal to the sum of distances between all pairs of edges of G. We give bounds on W_e in terms of order and size. In particular we prove the asymptotically sharp upper bound for graphs of order n.     

Graphs with small boundary

For a pair of vertices x and y in a graph G, we denote by d_G(x,y) the distance between x and y in G. We call x a boundary vertex of y if x and y belong to the same component and d_G(y,v)?d_G(y,x) for each neighbor v of x in G. A boundary vertex of some vertex is simply called a boundary vertex, and the set of boundary vertices in G is called the boundary of G, and is denoted by B(G).In this paper, we investigate graphs with a small boundary. Since a pair of farthest vertices are boundary vertices, |B(G)|?2 for every connected graph G of order at least two. We characterize the graphs with boundary of order at most three. We cannot give a characterization of graphs with exactly four boundary vertices, but we prove that such graphs have minimum degree at most six. Finally, we give an upper bound to the minimum degree of a connected graph G in terms of |B(G)|.     

Some Defective Parameters in Graphs

Given a graph G and an integer k, a set S of vertices in G is k-sparse if S induces a graph with a maximum degree of at most k. Many parameters in graph theory are defined in terms of independent sets. Accordingly, their definitions can be expanded taking into account the notion of k-sparse sets. In this discussion, we examine several of those extensions. Similarly, S is k-dense if S induces a k-sparse graph in the complement of G. A partition of V(G) is a k-defective cocoloring if each part is k-sparse or k-dense. The minimum order of all k-defective cocolorings is the k-defective cochromatic number of G and denoted z _k (G). Analogous notions are defined similarly for k-defective coloring where V(G) is partitioned into k-sparse parts. We show the NP-hardness of computing maximum k-defective sets in planar graphs with maximum degree at most k + 1 and arbitrarily large girth. We explore the extension of Ramsey numbers to k-sparse and k-dense sets of vertices. Lastly, we discuss some bounds related to k-defective colorings and k-defective cocolorings.     

Edge-disjoint spanners in tori

A spanning subgraph S=(V,E^′) of a connected graph G=(V,E) is an (x+c)-spanner if for any pair of vertices u and v, d_S(u,v)≤d_G(u,v)+c where d_G and d_S are the usual distance functions in G and S, respectively. The parameter c is called the delay of the spanner. We study edge-disjoint spanners in graphs in multi-dimensional tori. We show that each two-dimensional torus has a set of two edge-disjoint spanners of delay approximately the size of the smaller dimension. Moreover, we show that this delay is close to the best possible. In three-dimensional tori, we find a set of three edge-disjoint spanners with delay approximately the sum of the sizes of the two smaller dimensions when all dimensions are of even size. Surprisingly, we also find a set of two edge-disjoint spanners in three-dimensional tori of constant delay. In d-dimensional tori, we show that for any k≤d/5, there is a set of k edge-disjoint spanners with delay depending only on k and the size of the smaller k dimensions.     

Disjoint long cycles in a graph

We prove that if G is a graph of order at least 2k with k ? 9 and the minimum degree of G is at least k + 1, then G contains two vertex-disjoint cycles of order at least k. Moreover, the condition on the minimum degree is sharp.     

On a conjecture of Fink and Jacobson concerning k-domination and k-dependence

A set D of vertices of a graph is k-dependent if every vertex of D is joined to at most k?1 vertices in D. Let β_k(G) be the maximum order of a k-dependent set in G. A set D of vertices of G is k-dominating if every vertex not in D is joined to at least k vertices of D. Let γ_k(G) be the minimum order of a k-dominating set in G. Here we prove the following conjecture of Fink and Jacobson: for any simple graph G and any positive integer k, γ_k(G) ≤ β_k(G).     

Contractible Small Subgraphs in k-connected Graphs

We prove that if G is highly connected, then either G contains a non-separating connected subgraph of order three or else G contains a small obstruction for the above conclusion. More precisely, we prove that if G is k-connected (with k ≥ 2), then G contains either a connected subgraph of order three whose contraction results in a k-connected graph (i.e., keeps the connectivity) or a subdivision of ${K_4^-}$ whose order is at most 6.     

Distance graphs with maximum chromatic number

The distance graph G(D) has the set of integers as vertices and two vertices are adjacent in G(D) if their difference is contained in the set D⊂Z. A conjecture of Zhu states that if the chromatic number of G(D) achieves its maximum value |D|+1 then the graph has a triangle. The conjecture is proven to be true if |D|?3. We prove that the chromatic number of a distance graph with D={a,b,c,d} is five only if either D={1,2,3,4k} or D={a,b,a+b,b-a}. This confirms a stronger version of Zhu's conjecture for |D|=4, namely, if the chromatic number achieves its maximum value then the graph contains K₄.     

Partitioning graphs into complete and empty graphs

Given integers j and k and a graph G, we consider partitions of the vertex set of G into j+k parts where j of these parts induce empty graphs and the remaining k induce cliques. If such a partition exists, we say G is a (j,k)-graph. For a fixed j and k we consider the maximum order n where every graph of order n is a (j,k)-graph. The split-chromatic number of G is the minimum j where G is a (j,j)-graph. Further, the cochromatic number is the minimum j+k where G is a (j,k)-graph. We examine some relations between cochromatic, split-chromatic and chromatic numbers. We also consider some computational questions related to chordal graphs and cographs.     

The Harary index of ordinary and generalized quasi-tree graphs

The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. The quasi-tree graph is a graph G in which there exists a vertex v∈V(G) such that G?v is a tree. In this paper, we presented the upper and lower bounds on the Harary index of all quasi-tree graphs of order n and characterized the corresponding extremal graphs. Moreover we defined the k-generalized quasi-tree graph to be a connected graph G with a subset V _k ?V(G) where |V _k |=k such that G?V _k is a tree. And we also determined the k-generalized quasi-tree graph of order n with maximal Harary index for all values of k and the extremal one with minimal Harary index for k=2.     

New families of hypohamiltonian graphs

We construct three new infinite families of hypohamiltonian graphs having respectively 3k+1 vertices (k?3), 3k vertices (k?5) and 5k vertices (k?4); in particular, we exhibit a hypohamiltonian graph of order 19 and a cubic hypohamiltonian graph of order 20, the existence of which was still in doubt. Using these families, we get a lower bound for the number of non-isomorphic hypohamiltonian graphs of order 3k and 5k. We also give an example of an infinite graph G having no two-way infinite hamiltonian path, but in which every vertex-deleted subgraph G - x has such a path.     

A characterization of maximal non-k-factor-critical graphs

A graph G of order p is k-factor-critical,where p and k are positive integers with the same parity, if the deletion of any set of k vertices results in a graph with a perfect matching. G is called maximal non-k-factor-critical if G is not k-factor-critical but G+e is k-factor-critical for every missing edge e∉E(G). A connected graph G with a perfect matching on 2n vertices is k-extendable, for 1?k?n-1, if for every matching M of size k in G there is a perfect matching in G containing all edges of M. G is called maximal non-k-extendable if G is not k-extendable but G+e is k-extendable for every missing edge e∉E(G) . A connected bipartite graph G with a bipartitioning set (X,Y) such that |X|=|Y|=n is maximal non-k-extendable bipartite if G is not k-extendable but G+xy is k-extendable for any edge xy∉E(G) with x∈X and y∈Y. A complete characterization of maximal non-k-factor-critical graphs, maximal non-k-extendable graphs and maximal non-k-extendable bipartite graphs is given.     

Edge fault tolerance analysis of super k-restricted connected networks

An edge cut X of a connected graph G is a k-restricted edge cut if G-X is disconnected and every component of G-X has at least k vertices. Additionally, if the deletion of a minimum k-restricted edge cut isolates a connected component of k vertices, then the graph is said to be super-λ_k. In this paper, several sufficient conditions yielding super-λ_k graphs are given in terms of the girth and the diameter.     

Covering the vertices of a graph with cycles of bounded length

Let c_k(G) be the minimum number of elementary cycles of length at most k necessary to cover the vertices of a given graph G. In this work, we bound c_k(G) by a function of the order of G and its independence number.     

Sets in Abelian groups with distinct sums of pairs

A subset S={s₁,…,sk} of an Abelian group G is called an St-set of size k if all sums of t different elements in S are distinct. Let s(G) denote the cardinality of the largest S₂-set in G. Let v(k) denote the order of the smallest Abelian group for which s(G)?k. In this article, bounds for s(G) are developed and v(k) is determined for k?15 by computing s(G) for Abelian groups of order up to 183 using exhaustive backtrack search with isomorph rejection.