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Proper‐walk connection number of graphs
Author(s) -
BangJensen Jørgen,
Bellitto Thomas,
Yeo Anders
Publication year - 2021
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/jgt.22609
Subject(s) - mathematics , combinatorics , graph , undirected graph , connection (principal bundle) , discrete mathematics , multiple edges , connectivity , geometry
This paper studies the problem of proper‐walk connection number: given an undirected connected graph, our aim is to colour its edges with as few colours as possible so that there exists a properly coloured walk between every pair of vertices of the graph, that is, a walk that does not use consecutively two edges of the same colour. The problem was already solved on several classes of graphs but still open in the general case. We establish that the problem can always be solved in polynomial time in the size of the graph and we provide a characterization of the graphs that can be properly connected with k colours for every possible value of k .