Geodesic
Total Colorings of Planar Graphs with Small Maximum Degree
Bulletin of the Malaysian Mathematical Society, Tome 36 (2013) no. 3 Cet article a éte moissonné depuis la source Bulletin of the Malaysian Mathematical Society website

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Let $G$ be a planar graph of maximum degree $\Delta$ and girth $g$, and there is an integer $t(>g)$ such that $G$ has no cycles of length from $g+1$ to $t$. Then the total chromatic number of $G$ is $\Delta+1$ if $(\Delta,g,t)\in\{(5,4,6),(4,4,17)\}$; or $\Delta=3$ and $(g,t)\in\{(5,13),(6,11),(7,11),$ $(8,10),(9,10)\}$, where each vertex is incident with at most one $g$-cycle.
Classification : 05C15 
@article{BMMS_2013_36_3_a20,
     author = {Bing Wang and Jian-Liang Wu and Si-Feng Tian},
     title = {Total {Colorings} of {Planar} {Graphs} with {Small} {Maximum} {Degree}},
     journal = {Bulletin of the Malaysian Mathematical Society},
     year = {2013},
     volume = {36},
     number = {3},
     url = {http://geodesic.mathdoc.fr/item/BMMS_2013_36_3_a20/}
}
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Bing Wang; Jian-Liang Wu; Si-Feng Tian. Total Colorings of Planar Graphs with Small Maximum Degree. Bulletin of the Malaysian Mathematical Society, Tome 36 (2013) no. 3. http://geodesic.mathdoc.fr/item/BMMS_2013_36_3_a20/