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Neighbor sum distinguishing total chromatic number of $$K_4$$-minor free graph. (English) Zbl 1371.05097
Summary: A $$k$$-total coloring of a graph $$G$$ is a mapping $$\phi: V(G) \cup E(G)\to\{1,2,\dots,k\}$$ such that no two adjacent or incident elements in $$V(G)\cup E(G)$$ receive the same color. Let $$f(v)$$ denote the sum of the color on the vertex $$v$$ and the colors on all edges incident with $$v$$. We say that $$\phi$$ is a $$k$$-neighbor sum distinguishing total coloring of $$G$$ if $$f(u)\neq f(v)$$ for each edge $$uv\in E(G)$$. Denote $$\chi''_\Sigma(G)$$ the smallest value $$k$$ in such a coloring of $$G$$. Pilśniak and Woźniak conjectured that for any simple graph with maximum degree $$\Delta(G)$$, $$\chi''_\Sigma(G)\leqslant\Delta(G)+3$$. In this paper, by using the famous Combinatorial Nullstellensatz, we prove that for $$K_4$$-minor free graph $$G$$ with $$\Delta(G)\geqslant5$$, $$\chi''_\Sigma(G)=\Delta(G)+1$$ if $$G$$ contains no two adjacent $$\Delta$$-vertices, otherwise, $$\chi''_\Sigma(G)=\Delta(G)+2$$.

##### MSC:
 05C15 Coloring of graphs and hypergraphs
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##### References:
 [1] Alon, N, Combinatorial nullstellensatz, Combin Probab Comput, 8, 7-29, (1999) · Zbl 0920.05026 [2] Bondy J, Murty U. Graph Theory with Applications. New York: North-Holland,1976 · Zbl 1226.05083 [3] Cheng, X; Huang, D; Wang, G; Wu, J, Neighbor sum distinguishing total colorings of planar graphs with maximum degree, Discrete Appl Math, 190-191, 34-41, (2015) · Zbl 1316.05041 [4] Ding, L; Wang, G; Yan, G, Neighbour sum distinguishing total colorings via the combinatorial nullstellensatz, Sci China Math, 57, 1875-1882, (2014) · Zbl 1303.05058 [5] Dong, A; Wang, G, Neighbor sum distinguishing total colorings of graphs with bounded maximum average degree, Acta Math Sin (Engl Ser), 30, 703-709, (2014) · Zbl 1408.05061 [6] Li, H; Ding, L; Liu, B; Wang, G, Neighbor sum distinguishing total colorings of planar graphs, J Comb Optim, 30, 675-688, (2015) · Zbl 1325.05083 [7] Li, H; Liu, B; Wang, G, Neighbor sum distinguishing total colorings of K4-minor free graphs, Front Math China, 8, 1351-1366, (2013) · Zbl 1306.05066 [8] Pilśniak M, Woźniak M. On the adjacent-vertex-distinguishing index by sums in total proper colorings. http://www.ii.ui.edu.pl/preMD/index.php,2011 [9] Przybylo, J, Neighbour sum distinguishing total colorings via the combinatorial nullstellensatz, Discrete Appl Math, 202, 163-173, (2016) · Zbl 1330.05074 [10] Qu, C; Wang, G; Wu, J; Yu, X, On the neighbour sum distinguishing total coloring of planar graphs, Theoret Comput Sci, 609, 162-170, (2016) · Zbl 1331.05084 [11] Qu, C; Wang, G; Yan, G; Yu, X, Neighbor sum distinguishing total choosability of planar graphs, J Comb Optim, 32, 906-916, (2016) · Zbl 1348.05082 [12] Wang, J; Ma, Q; Han, X, Neighbor sum distinguishing total colorings of triangle free planar graphs, Acta Math Sin (Engl Ser), 31, 216-224, (2015) · Zbl 1317.05065 [13] Wang, J; Ma, Q; Han, X; Wang, X, A proper total coloring distinguishing adjacent vertices by sums of planar graphs without intersecting triangles, J Comb Optim, 32, 626-638, (2016) · Zbl 1343.05066 [14] Yao, J; Shao, Z; Xu, C, Neighbor sum distinguishing total choosability of graphs with δ = 3, Adv Math (China), 45, 343-348, (2016) · Zbl 1363.05087 [15] Yao, J; Xu, C, Neighbour sum distinguishing total coloring of graphs with maximum degree 3 or 4, J Shandong Univ Nat Sci, 50, 9-13, (2015) [16] Yao, J; Yu, X; Wang, G; Xu, C, Neighbour sum (set) distinguishing total choosability of d-degenerate graphs, Graphs Combin, 32, 1611-1620, (2016) · Zbl 1342.05052
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