Frederic Havet ; Nagarajan Paramaguru ; Rathinaswamy Sampathkumar - Detection number of bipartite graphs and cubic graphs

dmtcs:642 - Discrete Mathematics & Theoretical Computer Science, December 31, 2014, Vol. 16 no. 3 - https://doi.org/10.46298/dmtcs.642
Detection number of bipartite graphs and cubic graphsArticle

Authors: Frederic Havet ORCID1; Nagarajan Paramaguru 2; Rathinaswamy Sampathkumar 3

  • 1 Combinatorics, Optimization and Algorithms for Telecommunications
  • 2 Mathematics Wing, Directorate of Distance Education [India]
  • 3 Mathematics Section [India]

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.


Volume: Vol. 16 no. 3
Published on: December 31, 2014
Accepted on: June 9, 2015
Submitted on: October 23, 2012
Keywords: [INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]
Funding:
    Source : OpenAIRE Graph
  • Funder: French National Research Agency (ANR); Code: ANR-09-BLAN-0373

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