ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory

Article in press


M. Naeem, M.K. Siddiqui, M. Bača, A. Semaničová-Feňovčíková, F. Ashraf


On edge $H$-irregularity strengths of some graphs


Discussiones Mathematicae Graph Theory

Received: 2018-11-02, Revised: 2019-03-25, Accepted: 2019-03-25,


For a graph $G$ an edge-covering of $G$ is a family of subgraphs $H_1, H_2, \dots,$ $ H_t$ such that each edge of $E(G)$ belongs to at least one of the subgraphs $H_i$, $i=1, 2, \dots, t$. In this case we say that $G$ admits an $(H_1, H_2, \dots, H_t)$-$($edge$)$ covering. An $H$-covering of graph $G$ is an $(H_1, H_2, \dots, H_t)$-$($edge$)$ covering in which every subgraph $H_i$ is isomorphic to a given graph $H$.<br>Let $G$ be a graph admitting $H$-covering. An edge $k$-labeling $\alpha:E(G)\to \{1,2,\dots, k\}$ is called an $H$-irregular edge $k$-labeling of the graph $G$ if for every two different subgraphs $H'$ and $H''$ isomorphic to $H$ their weights $wt_{\alpha}(H')$ and $wt_{\alpha}(H'')$ are distinct. The weight of a subgraph $H$ under an edge $k$-labeling $\alpha$ is the sum of labels of edges belonging to $H$. The edge $H$-irregularity strength of a graph $G$, denoted by ${\rm{ehs}}(G,H)$, is the smallest integer $k$ such that $G$ has an $H$-irregular edge $k$-labeling.<br>In this paper we determine the exact values of ${\rm{ehs}}(G,H)$ for prisms, antiprisms, triangular ladders, diagonal ladders, wheels and gear graphs. Moreover the subgraph $H$ is isomorphic to only $C_{4}$, $C_{3}$ and $K_{4}$.


$H$-irregular edge labeling, edge $H$-irregularity strength, prism, antiprism, triangular ladder, diagonal ladder, wheel, gear graph