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We propose two conjectures on the chromatic polynomial of a graph and show their validity for several classes of graphs. Our conjectures are stronger than an older conjecture of Bartels and Welsh [1].
We prove that the edges of every even graph G = G1 + G2 that is the join of two regular graphs Gi = (Vi,Ei) can be coloured with Δ(G) colours, whenever Δ(G) = Δ(G2) + |V(G1)|. The proof of this result yields a combinatorial algorithm to optimally colour...
We prove that the edges of every even graph G=G1+G2 that is the join of two regular graphs G1 and G2 can be coloured with Δ(G) colours, whenever Δ(G)=Δ(G1)+|V2|. The proof of this result together with the results in De Simone and Galluccio (J. Comb. Optim. 18:417–428, 2009) states that every even graph G that is the join of two regular graphs is Class 1. The proof yields an efficient combinatorial...
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