Colourfully panconnected subgraphs II
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Let G be a connected k-colourable graph of order n > k. A subgraph H of G is k-colourfully panconnected in G if there is a k-colouring of G such that the colours are close together in H, in two different senses (called variegated and panconnected) to be made precise. Let sk(G) denote the smallest number of edges in a spanning k-colourfully panconnected subgraph H of G. It is conjectured that sk(G) = n − 1 if k > 4 and G is not a circuit (a connected 2-regular graph) with length ≡ 1 (mod k). It is proved that sk(G) = n−1 if G contains no circuit with length ≡ 1 (mod k), and sk(G) 6 2n− k− 1 whenever k > 4.
[1] Richard H. Schelp,et al. A special k-coloring for a connected k-chromatic graph , 1997, Discret. Math..