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Journal Articles Discrete Mathematics Year : 2002

## Recognizing Knödel graphs

Johanne Cohen
Pierre Fraigniaud
• Function : Author
• PersonId : 835294
Cyril Gavoille

#### Abstract

Given a circulant digraph H=(V,A) of order n and of generators \gamma_0,...,\gamma_k , 0\leq \gamma_i \leq n-1 , i=0,...,k, the bipartite incident-graph of H is a bipartite graph G=(V_1,V_2,E) of order 2n where V_1=V_2=V, and, for any , and any , \{x_1,x_2\}\in E \Leftrightarrow (x_1,x_2)\in A \Leftrightarrow \exists i \in \{0,\dots,k\} \mid x_2 = x_1 + \gamma_i \pmod{n} . Knödel graphs and Fibonacci graphs are two types of such graphs. They correspond to \gamma_i=2^i-1, and \gamma_i=F(i+1)-1 , respectively. Both graphs have been extensively studied for the purpose of fast communications in networks, and they have deserved a lot of attention in this context. In this paper, we show that there exists a polynomial-time algorithm to recognize Knödel graphs, and that the same technique applies to Fibonacci graphs. The algorithm is based on a characterization of the cycles of length six in these graphs (bipartite incident-graphs of circulant digraphs always have cycles of length six). A consequence of our result is that none of the Knödel graphs are edge-transitive, apart those of 2^k-2 vertices. An open problem that arises in this field is to know whether a polynomial-time algorithm exists for any infinite family of bipartite incident-graphs of circulant digraphs indexed by their number of vertices.

Other [cs.OH]

### Dates and versions

inria-00100969 , version 1 (26-09-2006)

### Identifiers

• HAL Id : inria-00100969 , version 1

### Cite

Johanne Cohen, Pierre Fraigniaud, Cyril Gavoille. Recognizing Knödel graphs. Discrete Mathematics, 2002, 250, pp.41-62. ⟨inria-00100969⟩

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