# Subpancyclicity of line graphs and degree sums along paths

## Abstract

A graph is called subpancyclic if it contains a cycle of length $\ell$ for each $\ell$ between 3 and the circumference of the graph. We show that if $G$ is a connected graph on $n\ge 146$ vertices such that $d(u)+d(v)+d(x)+d(y)>(n+10/2)$ for all four vertices $u,v,x,y$ of any path $P=uvxy$ in $G$, then the line graph $L(G)$ is subpancyclic, unless $G$ is isomorphic to an exceptional graph. Moreover, we show that this result is best possible, even under the assumption that $L(G)$ is hamiltonian. This improves earlier sufficient conditions by a multiplicative factor rather than an additive constant.
Original language English 1453-1463 11 Discrete applied mathematics 154 06EX1521/9 https://doi.org/10.1016/j.dam.2005.05.039 Published - 2 Jun 2006

• MSC-05C45
• MSC-05C35

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