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- W2964291259 abstract "A graph is said to be {it total-colored} if all the edges and the vertices of the graph are colored. A total-coloring of a graph is a {it total monochromatically-connecting coloring} ({it TMC-coloring}, for short) if any two vertices of the graph are connected by a path whose edges and internal vertices on the path have the same color. For a connected graph $G$, the {it total monochromatic connection number}, denoted by $tmc(G)$, is defined as the maximum number of colors used in a TMC-coloring of $G$. Note that a TMC-coloring does not exist if $G$ is not connected, in which case we simply let $tmc(G)=0$. In this paper, we first characterize all graphs of order $n$ and size $m$ with $tmc(G)=3,4,5,6,m+n-2,m+n-3$ and $m+n-4$, respectively. Then we determine the threshold function for a random graph to have $tmc(G)geq f(n)$, where $f(n)$ is a function satisfying $1leq f(n)<frac{1}{2}n(n-1)+n$. Finally, we show that for a given connected graph $G$, and a positive integer $L$ with $Lleq m+n$, it is NP-complete to decide whether $tmc(G)geq L$." @default.
- W2964291259 created "2019-07-30" @default.
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- W2964291259 date "2018-01-01" @default.
- W2964291259 modified "2023-09-24" @default.
- W2964291259 title "More on total monochromatic connection of graphs." @default.
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