Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8903632 | European Journal of Combinatorics | 2018 | 15 Pages |
Abstract
We consider the following extension of the concept of adjacent strong edge colourings of graphs without isolated edges. Two distinct vertices which are at distance at most r in a graph are called r-adjacent. The least number of colours in a proper edge colouring of a graph G such that the sets of colours met by any r-adjacent vertices in G are distinct is called the r-adjacent strong chromatic index of G and denoted by Ïa,râ²(G). It has been conjectured that Ïa,1â²(G)â¤Î+2 if G is connected of maximum degree Îâ¥2 and non-isomorphic to C5, while Hatami proved that there is a constant C, Câ¤300, such that Ïa,1â²(G)â¤Î+C if Î>1020 [J. Combin. Theory Ser. B 95 (2005) 246-256]. We conjecture that a similar statement should hold for any r, i.e., that for each positive integer r there exist constants δ0 and C such that Ïa,râ²(G)â¤Î+C for every graph without an isolated edge and with minimum degree δâ¥Î´0, and argue that a lower bound on δ is unavoidable in such a case (for r>2). Using the probabilistic method we prove such an upper bound to hold for graphs with δâ¥ÏµÎ, for every r and any fixed εâ(0,1], i.e., in particular for regular graphs. We also support the conjecture by proving an upper bound Ïa,râ²(G)â¤(1+o(1))Î for graphs with δâ¥r+2.
Related Topics
Physical Sciences and Engineering
Mathematics
Discrete Mathematics and Combinatorics
Authors
Jakub PrzybyÅo,