Two players, Dominator and Staller, alternate choosing vertices of a graph G, one at a time, such that each chosen vertex enlarges the set of vertices dominated so far. The aim of the Dominator is to finish the game as soon as possible, while the aim of the Staller is just the opposite. The game domination number γg(G) is the number of vertices chosen when Dominator starts the game and both pla...

For a simple graph of maximum degree ∆, the complexity of computing the fractional total chromatic number is unknown. Kilakos and Reed proved it lies between ∆+1 and ∆+2, and so we can approximate it within 1. In this paper, we strengthen this by characterizing exactly those simple graphs with fractional total chromatic number ∆ + 2. This yields a simple linear-time algorithm to determine wheth...

Let G be a simple graph and (G,) denotes the number of proper vertex colourings of G with at most  colours, which is for a fixed graph G , a polynomial in  , which is called the chromatic polynomial of G . Using the chromatic polynomial of some specific graphs, we obtain the chromatic polynomials of some nanostars.

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A multicircuit is a multigraph whose underlying simple graph is a circuit (a connected 2-regular graph). In this pair of papers, it is proved that every multicircuit C has total choosability (i.e., list total chromatic number) ch00(C ) equal to its ordinary total chromatic number 00(C ). In the present paper, the kernel method is used to prove this for every multicircuit that