Abstract
A graph G is edge-L-colorable, if for a given edge assignment L={L(e):e∈E(G)}, there exits a proper edge-coloring φ of G such that φ(e)∈L(e) for all e∈E(G). If G is edge-L-colorable for every edge assignment L with |L(e)|≥k for e∈E(G), then G is said to be edge-k-choosable. In this paper, We investigate structural of planar graphs without intersecting 4-cycles and show that every planar graph without intersecting 4-cycles is edge-k-choosable, where \(k=\max\{7,\Delta(G)+1\}\).
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References
Häggkvist, R., Chetwynd, A.: Some upper bounds on the total and list chromatic numbers of multigraphs. J. Graph Theory 16, 503–516 (1992)
Bollobás, B., Harris, A.J.: List-colourings of graphs. Graph Comb. 1, 115–127 (1985)
Galvin, F.: The list chromatic index of a bipartite multigraph. J. Comb. Theory, Ser. B 63, 153–158 (1995)
Häggkvist, R., Janssen, J.: New bounds on the list-chromatic index of the complete graph and other simple graphs. Comb. Probab. Comput. 6, 295–313 (1997)
Borodin, O.V., Kostochka, A.V., Woodall, D.R.: List edge and list total colourings of multigraphs. J. Comb. Theory, Ser. B 71, 184–204 (1997)
Wang, W.F., Lih, K.W.: Choosability, edge choosability and total choosability of outerplanar graphs. Eur. J. Comb. 22, 71–78 (2001)
Wu, B.R., An, X.H.: A note on edge choosability and degeneracy of planar graphs. Lect. Notes Comput. Sci. 5573, 249–257 (2009)
Vizing, V.G.: On an estimate of the chromatic class of p-graph. Metod. Diskretn. Anal. 3, 25–30 (1964) (in Russian)
Harris, A.J.: Problems and conjectures in extremal graph theory. Ph.D. Dissertation, Cambridge University, UK, 1984
Juvan, M., Mohar, B., Škrekovski, R.: Graphs of degree 4 are 5-choosable. J. Graph Theory 32, 250–262 (1999)
Kostochka, A.V.: List edge chromatic number of graphs with large girth. Discrete Math. 101, 189–201 (1992)
Borodin, O.V.: An extension of Kotzig’s’ theorem and the list edge coloring of planar graphs. Mat. Zametki 48, 22–48 (1990) (in Russian)
Shen, Y., Zheng, G., He, W., Zhao, Y.: Structural properties and edge choosability of planar graphs without 4-cycles. Discrete Math. 308, 5789–5794 (2008)
Wang, W.F., Lih, K.W.: Structural properties and edge choosability of planar graphs without 6-cycles. Comb. Probab. Comput. 10, 267–276 (2001)
Wang, W.F., Lih, K.W.: Choosability and edge choosability of planar graphs without intersecting triangles. SIAM J. Discrete Math. 15, 538–545 (2002)
Wang, W.F., Lih, K.W.: Choosability and edge choosability of planar graphs without five cycles. Appl. Math. Lett. 15, 561–565 (2002)
Zhang, L., Baoyindureng: Edge choosability of planar graphs without small cycles. Discrete Math. 283, 289–293 (2004)
Hou, J.F., Liu, G.Z., Cai, J.S.: List Edge and List Total colorings of planar graphs without 4-cycles. Theor. Comput. Sci. 369, 250–255 (2006)
Liu, B., Hou, J.F., Liu, G.Z.: List Edge and List Total colorings of planar graphs without short cycles. Inf. Process. Lett. 108, 347–351 (2008)
Hou, J.F., Liu, G.Z., Cai, J.S.: Edge-choosability of planar graphs without adjacent triangles or 7-cycles. Discrete Math. 309, 77–84 (2009)
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This work was supported by the Nature Science Foundation of Shandong Province (Y2008A20), also was supported by the Scientific Research and Development Project of Shandong Provincial Education Department (TJY0706) and the Science and Technology Foundation of University of Jinan (XKY0705).
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Ma, Q., Wang, J., Cai, J. et al. A note on edge-choosability of planar graphs without intersecting 4-cycles. J. Appl. Math. Comput. 36, 367–372 (2011). https://doi.org/10.1007/s12190-010-0408-5
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DOI: https://doi.org/10.1007/s12190-010-0408-5