The complete graph on n vertices is denoted by K, and the graph on n vertices in which each pair of vertices is joined by exactly 2 edges is denoted by 2K,. An m-cycle is a sequence of m distinct vertices (u1, u2, . . . . u,) such that ui is adjacent to ui+ 1 and U, is adjacent to u1 . A spanning subgraph H of G is a subgraph for which V(H) = V(G). An i-factor of a graph G is a spanning subgrap...

Given a graph with costs on the edges, the power of a node is the maximum cost of an edge leaving it, and the power of the graph is the sum of the powers of the nodes of this graph. Motivated by applications in wireless multi-hop networks, we consider four fundamental problems under the power minimization criteria: the Min-Power b-Edge-Cover problem (MPb-EC) where the goal is to find a min-powe...

Given a graph G, and a spanning subgraph H of G, a circular q-backbone k-coloring of (G,H) is a proper k-coloring c of G such that q ≤ |c(u) − c(v)| ≤ k − q, for every edge uv ∈ E(H). The circular qbackbone chromatic number of (G,H), denoted by CBCq(G,H), is the minimum integer k for which there exists a circular q-backbone k-coloring of (G,H). The Four Color Theorem implies that whenever G is ...

The clique number of an undirected graphG is the maximum order of a complete subgraph of G and is a well-known lower bound for the chromatic number ofG. Every proper k-coloring of G may be viewed as a homomorphism (an edge-preserving vertex mapping) of G to the complete graph of order k. By considering homomorphisms of oriented graphs (digraphs without cycles of length at most 2), we get a natu...

Let G be a 2-connected graph on n vertices such that d(x)+ d(y)+ d(z)>/n for all triples of independent vertices x,y,z. We prove that every longest cycle in G is a dominating cycle unless G is a spanning subgraph of a graph belonging to one of four easily specified classes of graphs.

In 1989, Thomassen asked whether there is an integer-valued function f(k) such that every f(k)-connected graph admits a spanning, bipartite k-connected subgraph. In this paper we take a first, humble approach, showing the conjecture is true up to a log n factor.

A set X of vertices of a graph G is said to be 1-dependent if the subgraph of G induced by X has maximum degree one. The 1-dependent Ramsey number t1(l, m) is the smallest integer n such that for any 2-edge colouring (R, B) of Kn, the spanning subgraph B of Kn has a 1-dependent set of size l or the subgraph R has a 1-dependent set of size m. The 2-edge colouring (R, B) is a t1(l, m) Ramsey colo...

It is known that graphs on n vertices with minimum degree at least 3 have spanning trees with at least n/4 + 2 leaves and that this can be improved to (n + 4)/3 for cubic graphs without the diamond K4 − e as a subgraph. We generalize the second result by proving that every graph with minimum degree at least 3, without diamonds and certain subgraphs called blossoms, has a spanning tree with at l...

For a subset W of vertices of an undirected graph G, let S(W ) be the subgraph consisting of W , all edges incident to at least one vertex in W , and all vertices adjacent to at least one vertex in W . If S(W ) is a tree containing all the vertices of G, then we call it a spanning star tree of G. In this case W forms a weakly connected but strongly acyclic dominating set for G. We prove that fo...

We introduce a graphical representation of quantifier-free predicate calculus formulas and a new rule of inference which employs this representation. The new rule is an amalgamation of resolution and Prawitz analysis which we call path resolution. Path resolution allows Prawitz analysis of an arbitrary subgraph of the graph representing a formula. If such a subgraph is not large enough to demon...