A graph G is called (k,d)*-choosable if, for every list assignment L satisfying |L(v)| = k for all v ∈ V(G), there is an L-coloring of G such that each vertex of G has at most d neighbors colored with the same color as itself. Ko-Wei Lih et al. used the way of discharging to prove that every planar graph without 4-cycles and i-cycles for some i ∈ 5,6,7 is (3,1)*-choosable. In this paper, we show that if G is 2-connected, we may just use Euler’s formula and the graph’s structural properties to prove these results. Furthermore, for 2-connected planar graph G, we use the same way to prove that, if G has no 4-cycles, and the number of 5-cycles contained in G is at most , then G is (3,1)*-choosable; if G has no 5-cycles, and any planar embedding of G does not contain any adjacent 3-faces and adjacent 4-faces, then G is (3,1)*-choosable.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1304,
author = {Yongqiang Zhao and Wenjie He},
title = {The use of Euler's formula in (3,1)*-list coloring},
journal = {Discussiones Mathematicae Graph Theory},
volume = {26},
year = {2006},
pages = {91-101},
zbl = {1106.05042},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1304}
}
Yongqiang Zhao; Wenjie He. The use of Euler's formula in (3,1)*-list coloring. Discussiones Mathematicae Graph Theory, Tome 26 (2006) pp. 91-101. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1304/
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