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On the line graph of a graph with diameter 2
Faculty Author(s): Zhan, Mingquan
Student Author(s): -
Department: MATH
Publication: Discrete Mathematics
Year: 2021
Abstract: A graph G is pancyclic if it contains cycles of all possible lengths. A graph G is 1-hamiltonian if the removal of at most 1 vertices from G results in a hamiltonian graph. In Veldman (1988) Veldman showed that the line graph L(G) of a connected graph G with diameter at most 2 is hamiltonian. In this paper, we continue studying the line graph L(G) of a connected graph G with |E(G)|≥3 and diameter at most 2 and prove the following:(i) L(G) is pancyclic if and only if G is not a cycle of length 4 or 5, and G is not the Petersen graph.(ii) L(G) is 1-hamiltonian if and only if κ(L(G))≥3.
Link: On the line graph of a graph with diameter 2