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Neighbor full sum distinguishing total coloring of Halin graphs
- Source :
- AIMS Mathematics, Vol 7, Iss 4, Pp 6959-6970 (2022)
- Publication Year :
- 2022
- Publisher :
- AIMS Press, 2022.
-
Abstract
- Let $ f: V(G)\cup E(G)\rightarrow \{1, 2, \dots, k\} $ be a total $ k $ -coloring of $ G $. Define a weight function on total coloring as $ \phi(x) = f(x)+\sum\limits_{e\ni x}f(e)+\sum\limits_{y\in N(x)}f(y), $ where $ N(x) = \{y\in V(G)|xy\in E(G)\} $. If $ \phi(x)\neq \phi(y) $ for any edge $ xy\in E(G) $, then $ f $ is called a neighbor full sum distinguishing total $ k $ -coloring of $ G $. The smallest value $ k $ for which $ G $ has such a coloring is called the neighbor full sum distinguishing total chromatic number of $ G $ and denoted by fgndi $ _{\sum}(G) $. Suppose that $ H = T\cup C $ is a Halin graph, where $ T $ and $ C $ are called the characteristic tree and the adjoint cycle, respectively. Let $ V_0\subseteq V(H)\setminus V(C) $ and each vertex in $ V_0 $ is adjacent to some vertices on $ C $. In this paper, we prove that the neighbor full sum distinguishing total chromatic number of two types of Halin graphs are not more than three: (i) 3-regular Halin graphs and (ii) every vertex of $ V_0 $ of a Halin graph with degree at least 4. The above results support a conjecture that fgndi $ _{\sum}(G)\leq 3 $ for any connected graph $ G $ of order at least three (Chang et al., 2022).
Details
- Language :
- English
- ISSN :
- 24736988
- Volume :
- 7
- Issue :
- 4
- Database :
- Directory of Open Access Journals
- Journal :
- AIMS Mathematics
- Publication Type :
- Academic Journal
- Accession number :
- edsdoj.53f2ef9a79f4558b5c46c88be56f05f
- Document Type :
- article
- Full Text :
- https://doi.org/10.3934/math.2022386?viewType=HTML