Journal of Combinatorial Optimization

, Volume 36, Issue 1, pp 23–34

Neighbor sum distinguishing total coloring of graphs with bounded treewidth

• Miaomiao Han
• You Lu
• Rong Luo
• Zhengke Miao
Article

Abstract

A proper total k-coloring $$\phi$$ of a graph G is a mapping from $$V(G)\cup E(G)$$ to $$\{1,2,\dots , k\}$$ such that no adjacent or incident elements in $$V(G)\cup E(G)$$ receive the same color. Let $$m_{\phi }(v)$$ denote the sum of the colors on the edges incident with the vertex v and the color on v. A proper total k-coloring of G is called neighbor sum distinguishing if $$m_{\phi }(u)\not =m_{\phi }(v)$$ for each edge $$uv\in E(G).$$ Let $$\chi _{\Sigma }^t(G)$$ be the neighbor sum distinguishing total chromatic number of a graph G. Pilśniak and Woźniak conjectured that for any graph G, $$\chi _{\Sigma }^t(G)\le \Delta (G)+3$$. In this paper, we show that if G is a graph with treewidth $$\ell \ge 3$$ and $$\Delta (G)\ge 2\ell +3$$, then $$\chi _{\Sigma }^t(G)\le \Delta (G)+\ell -1$$. This upper bound confirms the conjecture for graphs with treewidth 3 and 4. Furthermore, when $$\ell =3$$ and $$\Delta \ge 9$$, we show that $$\Delta (G) + 1\le \chi _{\Sigma }^t(G)\le \Delta (G)+2$$ and characterize graphs with equalities.

Keywords

Total coloring Neighbor sum distinguishing Treewidth

References

1. Alon N (1999) Combinatorial nullstellensatz. Comb Probab Comput. 8:7–29
2. Bodlaender HL (1998) A partial $$k$$-arboretum of graphs with bounded treewidth. Theor Comput Sci 209:1–45
3. Bondy JA, Murty USR (2008) Graph theory. In: GTM, vol 244. Springer, BerlinGoogle Scholar
4. Bruhn H, Lang R, Stein M (2016) List edge-coloring and total coloring in graphs of low treewidth. J Graph Theory 81(3):272–282
5. Dong AJ, Wang GH (2014) Neighbor sum distinguishing total coloring of graphs with bounded maximum average degree. Acta Math Sin 30(4):703–709
6. Ding LH, Wang GH, Yang GY (2014) Neighbor sum distinguishing total coloring via the combinatorial Nullstellensatz. Sin China Ser Math 57(9):1875–1882
7. Kalkowski M (2009) A note on 1,2-conjecture, in Ph.D. ThesisGoogle Scholar
8. Kalkowski M, Karoński M, Pfender F (2010) Vertex coloring edge-weightings: towards the 1–2–3-conjecture. J Comb Theory Ser B 100:347–349
9. Karoński M, Łuczak T, Thomason A (2004) Edge weights and vertex colours. J Comb Theory Ser B 91(1):151–157
10. Lang R (2015) On the list chromatic index of graphs of tree-width 3 and maximum degree 7. arXiv:1504.02122
11. Li HL, Liu BQ, Wang GH (2013) Neighbor sum distinguishing total coloring of $$K_4$$-minor-free graphs. Front Math China 8(6):1351–1366
12. Li HL, Ding LH, Liu BQ, Wang GH (2015) Neighbor sum distinguishing total colorings of planar graphs. J Comb Optim 30(3):675–688
13. Lu Y, Han M, Luo R (2018) Neighbor sum distinguishing total coloring and list neighbor sum distinguishing total coloring. Discrete Appl Math 237:109–115Google Scholar
14. Meeks K, Scott A (2016) The parameterised complexity of list problems on graphs of bounded treewidth. Inf Comput 251:91–103
15. Pilśniak M, Woźniak M (2015) On the total-neighbor distinguishing index by sums. Graphs Comb 31:771–782
16. Przybyło J, Woźniak M (2010) On a 1,2 conjecture. Discrete Math Theor Comput Sci 12(1):101–108
17. Yao JJ, Yu XW, Wang GH, Xu CQ (2016) Neighbor sum (set) distinguishing total choosability of $$d$$-degenerate graphs. Graphs Comb 32(4):1611–1620