Geodesic
A note on face coloring entire weightings of plane graphs
Discussiones Mathematicae. Graph Theory, Tome 34 (2014) no. 2, pp. 421-426.

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Given a weighting of all elements of a 2-connected plane graph G = (V,E, F), let f(α) denote the sum of the weights of the edges and vertices incident with the face α and also the weight of α. Such an entire weighting is a proper face colouring provided that f(α) ≠ f(β) for every two faces α and β sharing an edge. We show that for every 2-connected plane graph there is a proper face-colouring entire weighting with weights 1 through 4. For some families we improved 4 to 3.
Keywords: entire weighting, plane graph, face colouring 
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Jendrol, Stanislav; Šugerek, Peter. A note on face coloring entire weightings of plane graphs. Discussiones Mathematicae. Graph Theory, Tome 34 (2014) no. 2, pp. 421-426. http://geodesic.mathdoc.fr/item/DMGT_2014_34_2_a15/

[1] L. Addario-Berry, K. Dalal, C. McDiarmid, B.A. Reed and A. Thomason, Vertexcolouring edge-weightings, Combinatorica 27 (2007) 1-12. doi:10.1007/s00493-007-0041-6

[2] L. Addario-Berry, K. Dalal and B.A. Reed, Degree constrainted subgraphs, Discrete Appl. Math. 156 (2008) 1168-1174. doi:10.1016/j.dam.2007.05.059

[3] K. Appel and W. Haken, Every planar map is four-colorable, I. Discharging, Illinois J. Math. 21 (1977) 429-490.

[4] M. Bača, S. Jendrol’, K.M. Kathiresan and K. Muthugurupackiam, Entire labeling of plane graph, (submitted).

[5] M. Bača, S. Jendrol’, M. Miller and J. Ryan, On irregular total labellings, Discrete Math. 307 (2007) 1378-1388. doi:10.1016/j.disc.2005.11.075

[6] J.A. Bondy and U.S.R. Murty, Graph Theory (Springer-Verlag, Heidelberg, 2008).

[7] A.J. Dong and G.H. Wang, Neighbor sum distinguishing colorings of some graphs, Discrete Math. Algorithms Appl. (2012) 4(4) 1250047. doi:10.1142/S1793830912500474

[8] E. Flandrin, J.F. Saclé, A. Marczyk, J. Przybyło and M. Woźniak, Neighbor sum distinguishing index, Graphs Combin. 29 (2013) 1329-1336. doi:10.1007/s00373-012-1191-x

[9] A. Frieze, R.J. Gould, M. Karoński and F. Pfender, On graph irregularity strenght, J. Graph Theory 41 (2002) 120-137. doi:10.1002/jgt.10056

[10] H. Grötzsch, Zur Theorie der discreten Gebilde. VII. Ein Dreifarbensatz für dreikreisfreie Netze auf der Kugel., Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg, Math.-Mat. Reihe 8 (1958/1959) 109-120.

[11] G. Chartrand, M.S. Jacobson, L. Lehel, O.R. Oellermann, S. Ruiz and F. Saba, Irregular networks, Congr. Numer. 64 (1988) 187-192.

[12] M. Kalkowski, A note on 1, 2-conjecture, Electron. J. Combin. (to appear).

[13] M. Kalkowski, M. Karoński and F. Pfender, Vertex-coloring edge-weightings: towards the 1-2-3-conjecture, J. Combin. Theory (B) 100 (2010) 347-349. doi:10.1016/j.jctb.2009.06.002

[14] M. Karoński, T. Łuczak and A. Thomason, Edge weights and vertex colours, J. Combin. Theory (B) 91 (2004) 151-157. doi:10.1016/j.jctb.2003.12.001

[15] J. Przybyło and M. Woźniak, On 1, 2 conjecture, Discrete Math. Theor. Comput. Sci. 12 (2010) 101-108.

[16] T. Wang and Q. Yu, On vertex-coloring 13-edge-weighting, Front. Math. China 3 (2008) 1-7. doi:10.1007/s11464-008-0041-x

[17] W. Wang and X. Zhu, Entire colouring of plane graphs, J. Combin. Theory (B) 101 (2011) 490-501. doi:10.1016/j.jctb.2011.02.006