Xinmin Hou

On the total {k}-domination number of Cartesian products of graphs

AbstractLet γt{k}(G) denote the total {k}-domination number of graph G, and let

Forwarding indices of folded n -cubes

G\mathbin{\square}H

Group Connectivity of Complementary Graphs

denote the Cartesian product of graphs G and H. In this paper, we show that for any graphs G and H without isolated vertices,

Paired Domination Vertex Critical Graphs

\gamma _{t}^{\{k\}}(G)\gamma _{t}^{\{k\}}(H)\le k(k+1)\gamma _{t}^{\{k\}}(G\mathbin{\square}H)

On the (2,2)-domination number of trees

. As a corollary of this result, we have

Total domination of Cartesian products of graphs

\gamma _{t}(G)\gamma _{t}(H)\le 2\gamma _{t}(G\mathbin{\square}H)

On Perfect Matching Coverings and Even Subgraph Coverings

for all graphs G and H without isolated vertices, which is given by Pak Tung Ho (Util. Math., 2008, to appear) and first appeared as a conjecture proposed by Henning and Rall (Graph. Comb. 21:63–69, 2005).

A note on shortest cycle covers of cubic graphs

For a given connected graph G of order n, a routing R is a set of n(n - 1) elementary paths specified for every ordered pair of vertices in G. The vertex (resp. edge) forwarding index of a graph is the maximum number of paths of R passing through any vertex (resp. edge) in the graph. In this paper, the authors determine the vertex and the edge forwarding indices of a folded n-cube as (n-1)2n-1 +1-((n+l)/2) (n ⌊n+1/2⌋) and 2n-(n,⌊n+1/2⌋),respectively.

The total {k}-domatic number of wheels and complete graphs

Let G be a 2-edge-connected undirected graph, A be an (additive) abelian group and A* = A−{0}. A graph G is A-connected if G has an orientation D(G) such that for every function b: V(G)↦A satisfying , there is a function f: E(G)↦A* such that for each vertex v∈V(G), the total amount of f values on the edges directed out from v minus the total amount of f values on the edges directed into v equals b(v). For a 2-edge-connected graph G, define Λg(G) = min{k: for any abelian group A with |A|⩾k, G is A-connected }. In this article, we prove the following Ramsey type results on group connectivity: Let G be a simple graph on n⩾6 vertices. If min{δ(G), δ(Gc)}⩾2, then either Λg(G)⩽4, or Λg(Gc)⩽4. Let Z3 denote the cyclic group of order 3, and G be a simple graph on n⩾44 vertices. If min{δ(G), δ(Gc)}⩾4, then either G is Z3-connected, or Gc is Z3-connected.

Decomposition of Graphs into (k,r)-Fans and Single Edges: DECOMPOSITION OF GRAPHS INTO (K,R)-FANS AND SINGLE EDGES

Let γpr(G) denote the paired domination number of graph G. A graph G with no isolated vertex is paired domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, γpr (G – v) < γpr(G). We call these graphs γpr-critical. In this paper, we present a method of constructing γpr-critical graphs from smaller ones. Moreover, we show that the diameter of a γpr-critical graph is at most