Abstract
Let \(G=(V,E)\) be a graph with maximum degree \(\varDelta (G)\) and \(\phi :V\cup E\rightarrow \{1,2,\ldots ,k\}\) be a proper total coloring of the graph G. Let S(v) denote the set of the color on vertex v and the colors on the edges incident with v. Let f(v) denote the sum of the color on vertex v and the colors on the edges incident with v. The proper total coloring \(\phi \) is called neighbor set distinguishing or adjacent vertex distinguishing if \(S(u)\ne S(v)\) for each edge \(uv\in E(G)\). We say that \(\phi \) is neighbor sum distinguishing if \(f(u)\ne f(v)\) for each edge \(uv\in E(G)\). In both problems the challenging conjectures presume that such colorings exist for any graph G if \(k\ge \varDelta (G)+3\). Ding et al. proved in both problems \(k\ge \varDelta (G)+2d\) is sufficient for d-degenerate graph G. In this paper, we improve this bound and prove that \(k\ge \varDelta (G)+d+1\) is sufficient for d-degenerate graph G with \(d\le 8\) and \(\varDelta (G)\ge 2d\) or \(d\ge 9\) and \(\varDelta (G)\ge \frac{5}{2}d-5\). In fact, we prove these results in their list versions. As a consequence, we obtain an upper bound of the form \(\varDelta (G)+C\) for some families of graphs, e.g. \(\varDelta (G)+6\) for planar graphs with \(\varDelta (G)\ge 10\). In particular, we therefore obtain that when \(\varDelta (G)\ge 4\) two conjectures we mentioned above hold for 2-degenerate graphs in their list versions.
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References
Alon, N.: Combinatorial Nullstellensatz. Comb. Probab. Comput. 8, 7–29 (1999)
Bondy, J., Murty, U.: Graph theory. Springer, London (2008)
Chen, X.: On the adjacent vertex distinguishing total coloring numbers of graphs with \(\varDelta =3\). Discrete Math. 308(17), 4003–4007 (2008)
Cheng, X., Wu, J., Huang, D., Wang, G.: Neighbor sum distinguishing total colorings of planar with maximum degree \(\varDelta \). Discret. Appl. Math. 190, 34–41 (2015)
Ding, L., Wang, G., Wu, J., Yu, J.: Neighbor sum (set) distinguishing total choosability via the Combinatorial Nullstellensatz. Sci. China. Math. 57(9), 1875–1882
Ding, L., Wang, G., Yan, G.: Neighbor sum distinguishing total colorings via the Combinatorial Nullstellensatz. Sci. China Math. 57(9), 1875–1882 (2014)
Dong, A., Wang, G.: Neighbor sum distinguishing total coloring of graphs with bounded maximum average degree. Acta Math. Sin. 30(4), 703–709 (2014)
Li, H., Ding, L., Liu, B., Wang, G.: Neighbor sum distinguishing total colorings of planar graphs. J. Comb. Optim. 30(3), 675–685 (2015)
Li, H., Liu, B., Wang, G.: Neighbor sum distinguishing total colorings of \(K_{4}\)-minor free graphs. Front. Math. China 8(6), 1351–1366 (2013)
Przybyło, J.: Neighbor distinguishing edge colorings via Combinatorial Nullstellensatz. SIAM J. Discrete Math. 27(3), 1313–1322 (2013)
Pilśniak, M., Woźniak, M.: On the adjacent vertex distinguishing index by sums in total proper colorings. Graphs Comb. (2013). doi:10.1007/s00373-013-1399-4
Wang, G., Ding, L., Cheng X., Wu, J.: Improved bounds for neighbor sum (set) distinguishing choosability of planar graphs, Submitted
Wang, W., Huang, D.: The adjacent vertex distinguishing total coloring of planar graphs. J. Comb. Optim. 27(2), 379–396 (2014)
Wang, W., Wang, P.: On adjacent-vertex-distinguishing total coloring of \(K_{4}\)-minor free graphs. Sci. China Ser. A 39(12), 1462–1472 (2009)
Wang, Y., Wang, W.: Adjacent vertex distinguishing total colorings of outerplanar graphs. J. Comb. Optim. 19, 123–133 (2010)
Yao, J., Yu, X., Wang, G., Xu, C.: Neighbor sum distinguishing total coloring of 2-degenerate graph, Submitted
Zhang, Z., Chen, X., Li, J., Yao, B., Lu, X., Wang, J.: On adjacent-vertex-distinguishing total coloring of graphs. Sci. China Ser. A 48(3), 289–299 (2005)
Acknowledgments
This work was supported by the National Natural Science Foundation of China (11301134,11301135,11471193), the Natural Science Foundation of Hebei Province (A2015202301), the University Science and Technology Project of Hebei Province (ZD2015106) and the Scientific Research Foundation for the Excellent Middle-Aged and Young Scientists of Shandong Province (BS2012SF016).
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Yao, J., Yu, X., Wang, G. et al. Neighbor Sum (Set) Distinguishing Total Choosability of d-Degenerate Graphs. Graphs and Combinatorics 32, 1611–1620 (2016). https://doi.org/10.1007/s00373-015-1646-y
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DOI: https://doi.org/10.1007/s00373-015-1646-y
Keywords
- Neighbor sum (set) distinguishing total coloring
- Adjacent vertex distinguishing total coloring
- d-Degenerate graph
- List total coloring
- Combinatorial Nullstellensatz