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Detection number of bipartite graphs and cubic graphs

Frédéric Havet 1 Nagarajan Paramaguru 2 Rathinaswamy Sampathkumar 3 
1 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : For a connected graph G of order |V(G)| ≥3 and a k-labelling c : E(G) →{1,2,…,k} of the edges of G, the code of a vertex v of G is the ordered k-tuple (ℓ1,ℓ2,…,ℓk), where ℓi is the number of edges incident with v that are labelled i. The k-labelling c is detectable if every two adjacent vertices of G have distinct codes. The minimum positive integer k for which G has a detectable k-labelling is the detection number det(G) of G. In this paper, we show that it is NP-complete to decide if the detection number of a cubic graph is 2. We also show that the detection number of every bipartite graph of minimum degree at least 3 is at most 2. Finally, we give some sufficient condition for a cubic graph to have detection number 3.
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Frédéric Havet, Nagarajan Paramaguru, Rathinaswamy Sampathkumar. Detection number of bipartite graphs and cubic graphs. Discrete Mathematics and Theoretical Computer Science, 2014, Vol. 16 no. 3 (3), pp.333-342. ⟨10.46298/dmtcs.642⟩. ⟨hal-01110978⟩

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