### Abstract

Putting the concept of line graph in a more general setting, for a positive integer k the k-line graph L_{k}(G) of a graph G has the Kk-subgraphs of G as its vertices, and two vertices of L_{k}(G) are adjacent if the corresponding copies of K_{k} in G share k-1 vertices. Then, 2-line graph is just the line graph in usual sense, whilst 3-line graph is also known as triangle graph. The k-anti-Gallai graph Δ_{k}(G) of G is a specified subgraph of L_{k}(G) in which two vertices are adjacent if the corresponding two K_{k}-subgraphs are contained in a common K_{k+1}-subgraph in G. We give a unified characterization for nontrivial connected graphs G and F such that the Cartesian product G□F is a k-line graph. In particular for k = 3, this answers the question of Bagga (2004), yielding the necessary and suficient condition that G is the line graph of a triangle-free graph and F is a complete graph (or vice versa). We show that for any k ≥ 3, the k-line graph of a connected graph G is isomorphic to the line graph of G if and only if G = K_{k+2}. Furthermore, we prove that the recognition problem of k-line graphs and that of k-anti-Gallai graphs are NP-complete for each k ≥ 3.

Original language | English |
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Article number | #P3.33 |

Journal | Electronic Journal of Combinatorics |

Volume | 22 |

Issue number | 3 |

Publication status | Published - Sep 11 2015 |

### Keywords

- Anti-Gallai graph
- Cartesian product graph
- Triangle graph
- k-line graph

### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics
- Applied Mathematics

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## Cite this

*Electronic Journal of Combinatorics*,

*22*(3), [#P3.33].