^{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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