# Neighbor sum distinguishing total coloring of graphs with bounded treewidth

Neighbor sum distinguishing total coloring of graphs with bounded treewidth A proper total k-coloring $$\phi$$ ϕ of a graph G is a mapping from $$V(G)\cup E(G)$$ V ( G ) ∪ E ( G ) to $$\{1,2,\dots , k\}$$ { 1 , 2 , ⋯ , k } such that no adjacent or incident elements in $$V(G)\cup E(G)$$ V ( G ) ∪ E ( G ) receive the same color. Let $$m_{\phi }(v)$$ m ϕ ( 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 $$m_{\phi }(u)\not =m_{\phi }(v)$$ m ϕ ( u ) ≠ m ϕ ( v ) for each edge $$uv\in E(G).$$ u v ∈ E ( G ) . Let $$\chi _{\Sigma }^t(G)$$ χ Σ t ( G ) be the neighbor sum distinguishing total chromatic number of a graph G. Pilśniak and Woźniak conjectured that for any graph G, $$\chi _{\Sigma }^t(G)\le \Delta (G)+3$$ χ Σ t ( G ) ≤ Δ ( G ) + 3 . In this paper, we show that if G is a graph with treewidth $$\ell \ge 3$$ ℓ ≥ 3 and $$\Delta (G)\ge 2\ell +3$$ Δ ( G ) ≥ 2 ℓ + 3 , then $$\chi _{\Sigma }^t(G)\le \Delta (G)+\ell -1$$ χ Σ t ( G ) ≤ Δ ( G ) + ℓ - 1 . This upper bound confirms the conjecture for graphs with treewidth 3 and 4. Furthermore, when $$\ell =3$$ ℓ = 3 and $$\Delta \ge 9$$ Δ ≥ 9 , we show that $$\Delta (G) + 1\le \chi _{\Sigma }^t(G)\le \Delta (G)+2$$ Δ ( G ) + 1 ≤ χ Σ t ( G ) ≤ Δ ( G ) + 2 and characterize graphs with equalities. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Combinatorial Optimization Springer Journals

# Neighbor sum distinguishing total coloring of graphs with bounded treewidth

, Volume 36 (1) – Mar 23, 2018
12 pages

/lp/springer_journal/neighbor-sum-distinguishing-total-coloring-of-graphs-with-bounded-2LbOFqwvEH
Publisher
Springer US
Subject
Mathematics; Combinatorics; Convex and Discrete Geometry; Mathematical Modeling and Industrial Mathematics; Theory of Computation; Optimization; Operations Research/Decision Theory
ISSN
1382-6905
eISSN
1573-2886
D.O.I.
10.1007/s10878-018-0271-0
Publisher site
See Article on Publisher Site

### Abstract

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

### Journal

Journal of Combinatorial OptimizationSpringer Journals

Published: Mar 23, 2018

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