Total Colorings of Degenerated Graphs

Isobe, Shuji; Zhou, Xiao; Nishizeki, Takao

doi:10.1007/3-540-48224-5_42

Cited by 7 publications

(4 citation statements)

References 13 publications

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“…The total coloring has been intensively studied [17,15,3,2,4,9]. By our results, we extend in a certain sense the results of Borodin et al concerning the total coloring of planar graphs with large girth [4] and with large maximum degree [2]; by the way, we will use some techniques developed in these papers.…”

Section: Introductionsupporting

confidence: 54%

(d,1)-total labelling of planar graphs with large girth and high maximum degree

Bazzaro

Montassier

Raspaud

2007

Discrete Mathematics

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The (d, 1)-total number T d (G) of a graph G is the width of the smallest range of integers that suffices to label the vertices and the edges of G such that no two adjacent vertices have the same label, no two incident edges have the same label and the difference between the labels of a vertex and its incident edges is at least d. This notion was introduced in [F. Havet, (d, 1)-total labelling of graphs, in: Workshop on Graphs and Algorithms, Dijon, France, 2003]. In this paper, we prove that T d (G) + 2d − 2 for planar graphs with large girth and high maximum degree . Our results are optimal for d = 2.

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Section: Introductionsupporting

confidence: 54%

(d,1)-total labelling of planar graphs with large girth and high maximum degree

Bazzaro

Montassier

Raspaud

2007

Discrete Mathematics

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“…Hence the total coloring problem can be solved in time O(n log n) for a fairly large class of graphs including all planar graphs with sufficiently large maximum degree. We finally show that the total coloring problem can be solved in linear time for partial k-trees G with bounded k. Earlier versions of this paper were presented at [9,10].…”

Section: Introductionmentioning

confidence: 91%

Total Colorings Of Degenerate Graphs

2007

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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.

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“…2d þ 3 and MadðGÞ < If d ¼ 1, the ð1; 1Þ-total labeling is the traditional total coloring. The total coloring has been intensively studied [1,2,3,10,11,13]. To prove this theorem, we use some similar approaches to these developed in [3] and [1].…”

Section: Introductionmentioning

confidence: 97%