A note on face coloring entire weightings of plane graphs

• Autorzy: Stanislav Jendrol , Peter Šugerek
• Języki publikacji: EN

Abstrakty

EN

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 _ and also the weight of _. Such an entire weighting is a proper face colouring provided that f(α) ≠ f(β) for every two faces α and _ 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

Słowa kluczowe

EN

entire weighting; plane graph; face colouring

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2014
• Tom: 34
• Numer: 2
• Strony: 421-426

Daty

wydano

2014-05-01

online

2014-04-12

Twórcy

autor

Stanislav Jendrol

• Institute of Mathematics, Faculty of Science, Pavol Jozef Šafárik University, Jesenná 5, 040 01 Košice, Slovakia

autor

Peter Šugerek

• Institute of Mathematics, Faculty of Science, Pavol Jozef Šafárik University, Jesenná 5, 040 01 Košice, Slovakia

Bibliografia

• [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[Crossref]
• [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[Crossref]
• [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[Crossref]
• [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[Crossref]
• [8] E. Flandrin, J.F. Saclé, A. Marczyk, J. Przyby lo and M. Woźniak, Neighbor sum distinguishing index, Graphs Combin. 29 (2013) 1329-1336. doi:10.1007/s00373-012-1191-x[Crossref]
• [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[Crossref]
• [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[Crossref][WoS]
• [14] M. Karo´nski, T. Luczak and A. Thomason, Edge weights and vertex colours, J. Combin. Theory (B) 91 (2004) 151-157. doi:10.1016/j.jctb.2003.12.001[Crossref]
• [15] J. Przyby lo 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[WoS][Crossref]
• [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 [Crossref]

Identyfikatory

DOI

10.7151/dmgt.1738