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## Discussiones Mathematicae Graph Theory

2006 | 26 | 1 | 91-101
Tytuł artykułu

### The use of Euler's formula in (3,1)*-list coloring

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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 $11 + ⎣∑_{i≥5} [(5i-24)/4] |V_i|⎦$, 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.
Słowa kluczowe
EN
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
91-101
Opis fizyczny
Daty
wydano
2006
otrzymano
2005-01-18
poprawiono
2005-11-14
Twórcy
autor
• Department of Mathematics, Shijiazhuang College, Shijiazhuang 050801, P.R. China
autor
• Applied Mathematics Institute, Hebei University of Technology, Tianjin 300130, P.R. China
Bibliografia
• [1] N. Eaton and T. Hull, Defective list colorings of planar graphs, Bull. of the ICA 25 (1999) 79-87.
• [2] P. Erdös, A.L. Rubin and H. Taylor, Choosability in graphs, Congr. Numer. 26 (1979) 125-157.
• [3] K. Lih, Z. Song, W. Wang and K. Zhang, A note on list improper coloring planar graphs, Appl. Math. Letters 14 (2001) 269-273, doi: 10.1016/S0893-9659(00)00147-6.
• [4] R. Skrekovski, A grötzsch-type theorem for list colorings with impropriety one, Comb. Prob. Comp. 8 (1999) 493-507, doi: 10.1017/S096354839900396X.
• [5] R. Skrekovski, List improper colorings of planar graphs, Comb. Prob. Comp. 8 (1999) 293-299, doi: 10.1017/S0963548399003752.
• [6] R. Skrekovski, List improper colorings of planar graphs with prescribed girth, Discrete Math. 214 (2000) 221-233, doi: 10.1016/S0012-365X(99)00145-4.
• [7] C. Thomassen, 3-list coloring planar graphs of girth 5, J. Combin. Theory (B) 64 (1995) 101-107, doi: 10.1006/jctb.1995.1027.
• [8] V.G. Vizing, Vertex coloring with given colors (in Russian), Diskret. Analiz. 29 (1976) 3-10.
• [9] M. Voigt, A not 3-choosable planar graph without 3-cycles, Discrete Math. 146 (1995) 325-328, doi: 10.1016/0012-365X(94)00180-9.
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Bibliografia
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