On the inclusion chromatic index of a graph

Abstract

Let χ′⊂(G) be the least number of colours necessary to properly colour the edges of a graph G with minimum degree δgoodbreakinfix≥2 so that the set of colours incident with any vertex is not contained in a set of colours incident to any of its neighbours. We provide an infinite family of examples of graphs G with χ′⊂MathClass-open(GMathClass-close)≥true(1+1δ−1true)Δ, where normalΔ is the maximum degree of G, and we conjecture that χ′⊂ MathClass-open( G MathClass-close) ≤ true⌈ true( 1 + 1 δ − 1 true) Δ true⌉ for every connected graph with δ goodbreakinfix≥ 2 which is not isomorphic to C 5. The equality here is attained, for example, for the family of complete bipartite graphs. Using a probabilistic argument we support this conjecture by proving that χ ′ ⊂ MathClass-open( G MathClass-close) ≤ true( 1 + 4 δ true) Δ MathClass-open( 1 + o MathClass-open( 1 MathClass-close) MathClass-close) (for normalΔ goodbreakinfix→ ∞) for any fixed δ goodbreakinfix≥ 2, which implies that χ ′ ⊂ MathClass-open( G MathClass-close) ≤ true( 1 + 4 δ − 1 true) Δ for normalΔ large enough.

Inclusion chromatic index of random graphs

“…The inclusion-free colourings were introduced in [15], and independently in [11], where the authors proved that the general upper bound for χ G ′ ( ) ⊂ must depend not only on the maximum degree but also on the minimum degree. They also posed a conjecture that χ G ′ ( )…”

“…The inclusion‐free colourings were introduced in [15], and independently in [11], where the authors proved that the general upper bound for $${\chi }_{\subset }^{^{\prime} }(G)$$ must depend not only on the maximum degree but also on the minimum degree. They also posed a conjecture that $${\chi }_{\subset }^{^{\prime} }(G)\le \unicode{x02308}\left(1+\frac{1}{\delta -1}\right){\rm{\Delta }}\unicode{x02309}$$, and proved that $${\chi }_{\subset }^{^{\prime} }(G)\le \left(1+\frac{4}{\delta -1}\right){\rm{\Delta }}$$ for $${\rm{\Delta }}$$ large enough.…”