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In a paper by Zhang and Chen et al.(see [11]), a conjecture was made concerning the minimum number of colors χ at (G) required in a proper total-coloring of G so that any two adjacent vertices have different color sets, where the color set of a vertex ν is the set composed of the color of ν and the colors incident to ν. We find the exact values of χ at (G) and thus verify the conjecture when G is a Generalized Halin graph with maximum degree at least 6. A generalized Halin graph is a 2-connected plane graph G such that removing all the edges of the boundary of the exterior face of G (the degrees of the vertices in the boundary of exterior face of G are all three) gives a tree.
Acta Mathematicae Applicatae Sinica – Springer Journals
Published: Mar 13, 2008
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