On label graphoidal covering number-I

Document Type: Research Paper

Authors

1 DEPARTMENT OF MATHEMATICS THE MADURA COLLEGE MADURAI, TAMIL NADU

2 Department of Mathematics Thiagarajar Engineering College Madurai

Abstract

Let $G=(V‎, ‎E)$ be a graph with $p$ vertices and $q$ edges‎. ‎An acyclic‎ ‎graphoidal cover of $G$ is a collection $\psi$ of paths in $G$‎ ‎which are internally-disjoint and cover each edge of the graph‎ ‎exactly once‎. ‎Let $f‎: ‎V\rightarrow \{1‎, ‎2‎, ‎\ldots‎, ‎p\}$ be a bijective‎ ‎labeling of the vertices of $G$‎. ‎Let $\uparrow\!G_f$ be the‎ ‎directed graph obtained by orienting the edges $uv$ of $G$ from‎ ‎$u$ to $v$ provided $f(u)< f(v)$‎. ‎If the set $\psi_f$ of all‎ ‎maximal directed paths in $\uparrow\!G_f$‎, ‎with directions‎ ‎ignored‎, ‎is an acyclic graphoidal cover of $G$‎, ‎then $f$ is called‎ ‎a \emph{graphoidal labeling} of $G$ and $G$ is called a‎ ‎label graphoidal graph and $\eta_l=\min\{|\psi_f|‎: ‎f \ {\rm ‎is\ a\ graphoidal\ labeling\ of}\ G\}$ is called the label‎ ‎graphoidal covering number of $G$‎. ‎In this paper we characterize‎ ‎graphs for which (i) $\eta_l=q-m$‎, ‎where $m$ is the number of‎ ‎vertices of degree 2 and (ii) $\eta_l= q$‎. ‎Also‎, ‎we determine the value of label graphoidal covering number for unicyclic graphs‎.

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