JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2021, Vol. 56 ›› Issue (11): 76-82.doi: 10.6040/j.issn.1671-9352.0.2020.362

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Vertex-distinguishing Ⅰ-total colorings and vertex-distinguishing Ⅵ-total colorings of mC7

YANG Han, CHEN Xiang-en*   

  1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China
  • Published:2021-11-15

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Abstract: By using the methods of constructing a matrix which was composed of color sets and empty set as the elements, distributing color sets in advance and coloring explicitly, to give the optimal vertex distinguishing Ⅰ-total colorings and the optimal vertex distinguishing Ⅵ-total colorings of mC7. Thus vertex distinguishing Ⅰ-total chromatic numbers and the vertex distinguishing Ⅵ-total chromatic numbers of mC7 are determined. The results show that the VDITC conjecture and VDVITC conjecture are valid for mC7.

Key words: cycle, disjoint union, vertex-distinguishing Ⅰ-total coloring, vertex-distinguishing Ⅵ-total coloring, vertex-distinguishing Ⅰ-total chromatic number, vertex-distinguishing Ⅵ-total chromatic number

CLC Number: 

  • O157.5
[1] BURRIS A C. Vertex-distinguishing edge-coloring[D]. Memphis: Memphis State University, 1993.
[2] BURRIS A C, SCHELP R H. Vertex-distinguishing proper edge-colorings[J]. Journal of Graph Theory, 1997, 26(2):73-82.
[3] HORÑÁK M, SOTÁK R. Observability of complete multipartite graphs with equipotent parts[J]. Ars Combinatoria-Waterloo then Winnipeg-, 1995, 41(6):289-301.
[4] CERNY J, HORNAK M, SOTAK R. Observability of a graph[J]. Mathematica Slovaca, 1996, 46(1):21-31.
[5] BAZGAN C, HARKAT-BENHAMDINE A, LI H, et al. On the vertex-distinguishing proper edge-colorings of graphs[J]. Journal of Combinatorial Theory, Series B, 1999, 75(2):288-301.
[6] BALISTER P N, RIORDAN O M, SCHELP R H. Vertex-distinguishing edge colorings of graphs[J]. Journal of Graph Theory, 2003, 42(2):95-109.
[7] HARARY F, PLANTHOLT M. The point-distinguishing chromatic index[M]. New York: Wiley Inter Science, 1985.
[8] HORNÑÁK M, SOTAK R. The fifth jump of the point-distinguishing chromatic index of Kn,n[J]. Ars Combinatoria-Waterloo then Winnipeg-, 1996, 42:233-242.
[9] HORÑÁK M, SOTÁK R. Localization of jumps of the point-distinguishing chromatic index of Kn,n[J]. Discussiones Mathematicae Graph Theory, 1997, 17(2):243.
[10] HORÑÁK M, SALVI N Z. On the point-distinguishing chromatic index of complete bipartite graphs[J]. Ars Combinatoria-Waterloo then Winnipeg-, 2006, 80:75-85.
[11] Zagaglia Salvi N. On the point-distinguishing chromatic index of Kn,n[J]. Ars Combinatoria, 1988, 25B:93-104.
[12] SALVI N Z. On the value of the point-distinguishing chromatic index of Kn,n[J]. Ars Combinatoria, 1990, 29B:235-244.
[13] ZHANG Zhongfu, QIU Pengxiang, XU Baogen, et al. Vertex-distinguishing total colorings of graphs[J]. Ars Combinatoria, 2008, 87:33-45.
[14] 陈祥恩.图的可区别染色引论[M].北京:中国科学技术出版社, 2015:108-141 CHEN Xiangen. An introduction to the distinguish coloring of graphs[M]. Beijing: Science and Technology of China Press, 2015: 108-141.
[15] CHEN Xiangen, LI Zepeng. Vertex-distinguishing Ⅰ-total colorings of graphs[J]. Utilitas Mathematica, 2014, 95:319-327.
[16] CHEN X E, GAO Y P, YAO B. Not necessarily proper total colourings which are adjacent vertex distinguishing[J]. International Journal of Computer Mathematics, 2013, 90(11):2298-2307.
[17] 陈祥恩, 苗婷婷, 王治文. 两条路的联图的点可区别Ⅰ-全染色[J].山东大学学报(理学版), 2017, 52(4):30-33. CHEN Xiangen, MIAO Tingting, WANG Zhiwen. Vertex-distinguishing Ⅰ-total colorings of the join of two paths[J]. Journal of Shandong University(Natural Science), 2017, 52(4):30-33.
[18] 苗婷婷, 陈祥恩, 王治文. 圈与路的联图的点可区别Ⅰ-全染色和点可区别Ⅵ-全染色[J]. 大连理工大学学报(自然科学版), 2017, 57(4):430-453. MIAO Tingting, CHEN Xiangen, WANG Zhiwen. Vertex-distinguishing Ⅰ-total colorings and vertex-distinguishing Ⅵ-total colorings of join-graph of cycle and path[J]. Journal of Dalian University of Technology, 2017, 57(4):430-453.
[19] 苗婷婷, 陈祥恩, 王治文. Cm∨Cn,Cm∨Wn,Cm∨Fn的点可区别Ⅰ-全染色和点可区别Ⅵ-全染色[J]. 厦门大学学报(自然科学版), 2017, 56(6):870-875. MIAO Tingting, CHEN Xiangen, WANG Zhiwen. Vertex-distinguishing Ⅰ-total colorings and vertex-distinguishing Ⅵ-total colorings of Cm∨Cn,Cm∨Wn,Cm∨Fn[J]. Journal of Xiamen University(Natural Science), 2017, 56(6):870-875.
[20] 辛小青, 陈祥恩. m个点不交的C4的并的点可区别全染色[J]. 山东大学学报(理学版), 2010, 45(10):35-39,44. XIN Xiaoqing, CHEN Xiangen. Vertex distinguishing total chromatic number of mC4[J]. Journal of Shandong University(Natural Science), 2010, 45(10):35-39,44.
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