Let c be a proper k-coloring of a connected graph G. Let Π = {S1, S2, . . . , Sk} be the induced partition of V (G) by c, where Si is the partition class having all vertices with color i. The color code cΠ(v) of vertex v is the ordered k-tuple (d(v, S1), d(v, S2), . . . , d(v, Sk)), where d(v, Si) = min{d(v, x)|x ∈ Si}, for 1 ≤ i ≤ k. If all vertices of G have distinct color codes, then c is ca...

For any vertex v and any edge e in a non-trivial connected graph G, the distance sum d(v) of v is d(v) = ∑ u∈V d(v, u), the vertex-to-edge distance sum d1(v) of v is d1(v) = ∑ e∈E d(v, e), the edge-to-vertex distance sum d2(e) of e is d2(e) = ∑ v∈V d(e, v) and the edge-to-edge distance sum d3(e) of e is d3(e) = ∑ f∈E d(e, f). The set M(G) of all vertices v for which d(v) is minimum is the media...

A k-tree is a tree with maximum degree at most k. In this paper, we give a degree sum condition for a graph to have a spanning k-tree in which specified vertices have degree less than k. We denote by σk(G) the minimum value of the degree sum of k independent vertices in a graph G. Let k ≥ 3 and s ≥ 0 be integers, and suppose G is a connected graph and σk(G) ≥ |V (G)|+s−1. Then for any s specifi...

The average distance μ(G) of a connected graph G of order n is the average of the distances between all pairs of vertices of G, i.e., μ(G) = ( n 2 )−1∑ {x,y}⊂V (G) dG(x, y), where V (G) denotes the vertex set of G and dG(x, y) is the distance between x and y. We prove that if G is a κ-vertex-connected graph, κ ≥ 3 an odd integer, of order n, then μ(G) ≤ n 2(κ+1) + O(1). Our bound is shown to be...

We present a new, single-rate method for compressing the connectivity information of a connected 2-manifold triangle mesh with or without boundary. Traditional compression schemes interleave geometry and connectivity coding, and are thus typically unable to utilise information from vertices (mesh regions) they have not yet processed. With the advent of competitive point cloud compression scheme...

The vertex connectivity k of a graph is the smallest number of vertices whose deletion separates the graph or makes it trivial. We present the fastest known deterministic algorithm for finding the vertex connectivity and a corresponding Ž 3 separator. The time for a digraph having n vertices and m edges is O min k q 4 . n, k n m ; for an undirected graph the term m can be replaced by k n. A ran...

It is known that a 3-edge-connected graph has a spanning even subgraph in which every component contains at lest five vertices, and the lower bound is best possible. A natural question arises whether we can improve the lower bound by changing the spanning property with the dominating property. In this paper, we show that a 3-edge-connected cubic graph has a dominating even subgraph in which eve...

Suppose that Γ is a connected graph and G is a subgroup of the automorphism group Aut(Γ) of Γ. Then Γ is G-symmetric if G acts transitively on the arcs (and so the vertices) of Γ and Γ is G-semisymmetric if G acts edge transitively but not vertex transitively on Γ. If Γ is Aut(Γ)symmetric, respectively, Aut(Γ)-semisymmetric, then we say that Γ is symmetric, respectively, semisymmetric. If Γ is ...

The exchanged hypercube EH(s, t), proposed by Loh et al. [The exchanged hypercube, IEEE Transactions on Parallel and Distributed Systems 16 (9) (2005) 866-874], is obtained by removing edges from a hypercube Qs+t+1. This paper considers a kind of generalized measures κ(h) and λ(h) of fault tolerance in EH(s, t) with 1 6 s 6 t and determines κ(h)(EH(s, t)) = λ(h)(EH(s, t)) = 2(s+ 1− h) for any h...

A DD2-pair of a graph G is a pair (D,D2) of disjoint sets of vertices of G such that D is a dominating set and D2 is a 2-dominating set of G. Although there are infinitely many graphs that do not contain a DD2-pair, we show that every graph with minimum degree at least two has a DD2-pair. We provide a constructive characterization of trees that have a DD2-pair and show that K3,3 is the only con...