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All Journal Electronic Journal of Graph Theory and Applications (EJGTA)
Shariefuddin Pirzada
Department of Mathematics, University of Kashmir, Srinagar, Kashmir, India
Electrical & Electronics Engineering

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On 2-power unicyclic cubic graphs Shariefuddin Pirzada; Mushtaq Shah; Edy Tri Baskoro
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 10, No 1 (2022): Electronic Journal of Graph Theory and Applications
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2022.10.1.24

Abstract

In a graph, a cycle whose length is a power of two (that is, 2k) is called a 2-power cycle. In this paper, we show that the existence of an infinite family of cubic graphs which contain only one cycle whose length is a power of 2. Such graphs are called as 2-power unicyclic cubic graphs. Further we observe that the only 2-power cycle in a cubic graph cannot be removed implying that there does not exist a counter example for Erdos-Gyárfás conjecture.
On some covering graphs of a graph Shariefuddin Pirzada; Hilal A Ganie; Merajuddin Siddique
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 4, No 2 (2016): Electronic Journal of Graph Theory and Applications
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2016.4.2.2

Abstract

For a graph $G$ with vertex set $V(G)=\{v_1, v_2, \dots, v_n\}$, let $S$ be the covering set of $G$ having the maximum degree over all the minimum covering sets of $G$. Let $N_S[v]=\{u\in S : uv \in E(G) \}\cup \{v\}$ be the closed neighbourhood of the vertex $v$ with respect to $S.$ We define a square matrix $A_S(G)= (a_{ij}),$ by $a_{ij}=1,$ if $\left |N_S[v_i]\cap N_S[v_j] \right| \geq 1, i\neq j$ and 0, otherwise. The graph $G^S$ associated with the matrix $A_S(G)$ is called the maximum degree minimum covering graph (MDMC-graph) of the graph $G$. In this paper, we give conditions for the graph $G^S$ to be bipartite and Hamiltonian. Also we obtain a bound for the number of edges of the graph $G^S$ in terms of the structure of $G$. Further we obtain an upper bound for covering number (independence number) of $G^S$  in terms of the covering number (independence number) of $G$.
Co-Authors Edy Tri Baskoro Hilal A Ganie Merajuddin Siddique Mushtaq Shah