Vertex colouring and forbidden subgraphs - a survey

There is a great variety of colouring concepts and results in the literature. Here our focus is to survey results on vertex colourings of graphs defined in terms of forbidden induced subgraph conditions

Maclyn McCarty (1911-2005)

"If I have seen further, it is by standing on the shoulders of giants" (letter of Isaac Newton to Robert Hooke). This well-known sentence of Newton finds its correct meaning in biology through the work of Oswald Avery (1877-1955), Colin MacLeod (1909-1972), and Maclyn McCarty (1911-2005) that was published in 1944 in The Journal of Experimental Medicine, which showed that DNA carried genetic information. These giants of molecular biology attained scientific evidence to provide shoulders strong enough to allow Crick and Watson to build on the foundations laid down by this group to postulate, 9 years later, the double-helix model of DNA. Maclyn McCarty died in New York on January 3, 2005, at age 93. At the time of his death, he was an active editor of the above-mentioned journal, which is published by The Rockefeller University.Peer reviewe

An upper bound on the domination number of a graph with minimum degree two

AbstractA set S of vertices in a graph G is a dominating set of G if every vertex of V(G)∖S is adjacent to some vertex in S. The minimum cardinality of a dominating set of G is the domination number of G, denoted as γ(G). Let Pn and Cn denote a path and a cycle, respectively, on n vertices. Let k1(F) and k2(F) denote the number of components of a graph F that are isomorphic to a graph in the family {P3,P4,P5,C5} and {P1,P2}, respectively. Let L be the set of vertices of G of degree more than 2, and let G−L be the graph obtained from G by deleting the vertices in L and all edges incident with L. McCuaig and Shepherd [W. McCuaig, B. Shepherd, Domination in graphs with minimum degree two, J. Graph Theory 13 (1989) 749–762] showed that if G is a connected graph of order n≥8 with δ(G)≥2, then γ(G)≤2n/5, while Reed [B.A. Reed, Paths, stars and the number three, Combin. Probab. Comput. 5 (1996) 277–295] showed that if G is a graph of order n with δ(G)≥3, then γ(G)≤3n/8. As an application of Reed’s result, we show that if G is a graph of order n≥14 with δ(G)≥2, then γ(G)≤38n+18k1(G−L)+14k2(G−L)

Vertex Colouring and Forbidden Subgraphs - a Survey

There is a great variety of colouring concepts and results in the literature. Here our focus is to survey results on vertex colourings of graphs de ned in terms of forbidden induced subgraph conditions

Exact algorithms for MINIMUM DOMINATING SET

We design exact algorithms for several NP-complete graph-theoretic problems. The term exact algorithm is a shorthand for a fast exponential time algorithm. Our main result states that a minimum dominating set in a graph of order n can be found in time O(1.8899^n). The list of considered problems includes MINIMUM DOMINATING SET, 3-COLOURABILITY, MINIMUM CONNECTED DOMINATING SET (~MAXIMUM LEAF SPANNING TREE), MINIMUM EDGE DOMINATING SET, MINIMUM INDEPENDENT DOMINATING SET and MINIMUM MAXIMAL MATCHING