Neighbor full sum distinguishing total coloring of two types of Cartesian product graphs

YE Hongbo; YANG Chao; YIN Zhixiang; YAO Bing

doi:10.12299/jsues.21-0252

Volume 35 Issue 1

Sep. 2022

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Article Navigation > Journal of Shanghai University of Engineering Science > 2022 > 36(1): 91-97

YE Hongbo, YANG Chao, YIN Zhixiang, YAO Bing. Neighbor full sum distinguishing total coloring of two types of Cartesian product graphs[J]. Journal of Shanghai University of Engineering Science, 2022, 36(1): 91-97. doi: 10.12299/jsues.21-0252

Citation:

YE Hongbo, YANG Chao, YIN Zhixiang, YAO Bing. Neighbor full sum distinguishing total coloring of two types of Cartesian product graphs[J]. Journal of Shanghai University of Engineering Science, 2022, 36(1): 91-97. doi: 10.12299/jsues.21-0252

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Neighbor full sum distinguishing total coloring of two types of Cartesian product graphs

doi: 10.12299/jsues.21-0252

YE Hongbo^{1a
,},
YANG Chao^{1a, 1b
,
,},
YIN Zhixiang^{1a, 1b},
YAO Bing²

1a.
a. School of Mathematics, Physics and Statistics; b. Center of Intelligent Computing and Applied Statistics, Shanghai University of Engineering Science, Shanghai 201620, China
1b.
Center of Intelligent Computing and Applied Statistics, Shanghai University of Engineering Science, Shanghai 201620, China
2.
College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

Received Date: 2021-11-14
Publish Date: 2022-09-26

Abstract

Abstract

Let
\begin{document}$f:V(G) \cup E(G) \to \{ 1,2,\cdots ,k\}$\end{document}
be a proper
k-total coloring of a graph G. Define a weight function on total coloring as
$ \phi (x) = f(x) + \mathop \Sigma \limits_{e \mathrel\backepsilon x} f(e) + \mathop \Sigma \limits_{y \in N(x)} f(y) $
, where
$ N(x) = \{ y \in V(G)|xy \in E(G)\} $
. If
$ \phi (u) \ne \phi (v) $
for any edge
$ uv \in E(G) $
, then f is called a neighbor full sum distinguishing k-total coloring of G. The smallest value k for which G admins a neighbor full sum distinguishing total coloring with k colors is called the neighbor full sum distinguishing total chromatic number of G and denoted by
${\rm{ftnd}}{{\rm{i}}_\Sigma }(G)$
. The research conjectures that
${\rm{ftnd}}{{\rm{i}}_\Sigma }(G) \leqslant \Delta + 2$
for every graph except for
$ {K_2} $
, where ∆ represents the maximum degree of G. Meanwhile, we get this parameter for Cartesian product graphs of paths and paths, paths and cycles are ∆ + 1, respectively, which confirm the above conjecture.

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References(9)

References

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