Total Colorings Of Degenerate Graphs

Abstract

A total coloring of a graph G is a coloring of all elements of G, i.e. vertices and edges, such that no two adjacent or incident elements receive the same color. A graph G is sdegenerate for a positive integer s if G can be reduced to a trivial graph by successive removal of vertices with degree ≤ s. We prove that an s-degenerate graph G has a total coloring with ∆+1 colors if the maximum degree ∆ of G is sufficiently large, say ∆ ≥ 4s+3. Our proof yields an efficient algorithm to find such a total coloring. We also give a lineartime algorithm to find a total coloring of a graph G with the minimum number of colors if G is a partial k-tree, that is, the tree-width of G is bounded by a fixed integer k.

List Edge‐Coloring and Total Coloring in Graphs of Low Treewidth

“…Isobe et al [7] prove, with quite different methods, a very similar result: namely that every k-degenerate graph G with (G) ≥ 4k + 3 can be totally colored with (G) + 1 colors. So, the result of Isobe et al is at same time stronger and weaker, that is, their result covers more graphs but with a stricter requirement on the maximum degree.…”

“…Our argumentation so far is enough to prove that any graph of treewidth k and maximum degree (G) ≥ 3k − 1 satisfies χ (G) = (G) + 1. Indeed, note that with the assumption (G) ≥ 3k − 1, we obtain |M(w * )| ≥ 2k + 1 in (7). Plugging this into (6), and using (5), we get n α ≥ k + 1.…”

“…Isobe et al. prove, with quite different methods, a very similar result: namely that every k ‐degenerate graph G with can be totally colored with colors. So, the result of Isobe et al.…”

“…Setting δ = 1 if ρ x ∈ M(w * ) and δ = 0 otherwise, we deduce from Claim 9 and (4) that (7) ≥ k + 1.…”

Maximum cardinality neighbourly sets in quadrilateral free graphs

“…Hence the total time taken is O(n 2 ). In case of planar graphs the degree of the minimum degree vertex is at most five [8]. So each step will take O(1) time, resulting in an O(n) time algorithm for planar quadrilateral-free graphs.…”