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.
Complexity of determining the irregular chromatic index of a graph
1 Document citing this article
Source : OpenCitations
Bensmail, Julien, 2022, On The Hardness Of Determining The Irregularity Strength Of Graphs, Theoretical Computer Science, 937, pp. 96-107, 10.1016/j.tcs.2022.09.033.