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- W2964239655 abstract "In a given graph $G$, a set $S$ of vertices with an assignment of colors is a {sf defining set of the vertex coloring of $G$}, if there exists a unique extension of the colors of $S$ to a $Cchi(G)$-coloring of the vertices of $G$. A defining set with minimum cardinality is called a {sf smallest defining set} (of vertex coloring) and its cardinality, the {sf defining number}, is denoted by $d(G, Cchi)$. Let $ d(n, r, Cchi = k)$ be the smallest defining number of all $r$-regular $k$-chromatic graphs with $n$ vertices. Mahmoodian et. al cite{rkgraph} proved that, for a given $k$ and for all $n geq 3k$, if $r geq 2(k-1)$ then $d(n, r, Cchi = k)=k-1$. In this paper we show that for a given $k$ and for all $n < 3k$ and $rgeq 2(k-1)$, $d(n, r, Cchi=k)=k-1$." @default.
- W2964239655 created "2019-07-30" @default.
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- W2964239655 date "2009-01-01" @default.
- W2964239655 modified "2023-09-26" @default.
- W2964239655 title "Constructing regular graphs with smallest defining number." @default.
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