The 6-relaxed game chromatic number of outerplanar graphs

Jiaojiao Wu, Xuding Zhu
2008 Discrete Mathematics  
Suppose G is a graph and k, d are integers. The (k, d)-relaxed colouring game on G is a game played by two players, Alice and Bob, who take turns colouring the vertices of G with legal colours from a set X of k colours. Here a colour i is legal for an uncoloured vertex x if after colouring x with colour i, the subgraph induced by vertices of colour i has maximum degree at most d. Alice's goal is to have all the vertices coloured, and Bob's goal is the opposite: to have an uncoloured vertex
more » ... ut a legal colour. The d-relaxed game chromatic number of G, denoted by χ , is the least number k so that when playing the (k, d)-relaxed colouring game on G, Alice has a winning strategy. This paper proves that if G is an outerplanar graph, then χ (d) g (G) ≤ 2 for d ≥ 6.
doi:10.1016/j.disc.2007.11.015 fatcat:xm5pe667jbdrrlvfemmeznrmji