## Abstract

The quadrilateral line graph Q(G) of a graph G is that graph whose vertices are the edges of G and such that two vertices of Q(G) are adjacent if and only if the corresponding edges of G are adjacent and belong to a common 4-cycle. It is shown that if G is a graph every edge of which belongs to a 4-cycle and neither K_{2,3} nor K_{4} is a subgraph of G, then every component of Q(G) is eulerian. Also, if G is a connected graph of order at least 4, every two adjacent edges of which belong to a common 4-cycle, then Q(G) is hamiltonian. For n ≥ 2, the nth iterated quadrilateral line graph Q^{n} (G) of a graph G is defined as Q(Q^{n-1}(G)), where Q^{1}(G) = Q(G) and Q^{n-1}(G) is assumed to be nonempty. It is shown that if G is a graph containing 4-cycles that contains no subgraph isomorphic to K_{1} + P_{4}, P_{3} × K_{2}, K_{2,3}, or K_{4}, then the sequence {Q^{k}(G)} converges to mC_{4} for some positive integer m.

Original language | English |
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Pages (from-to) | 3-17 |

Number of pages | 15 |

Journal | Utilitas Mathematica |

Volume | 50 |

Publication status | Published - 1996 |

Externally published | Yes |

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty
- Applied Mathematics