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Motivated by longstanding conjectures regarding decompositions of graphs into paths and cycles, we prove the following optimal decomposition results for random graphs. Let 0 < p < 1 be constant and let G ∼ G n , p . Let odd ( G ) be the number of odd degree vertices in G . Then a.a.s. the following hold: (i) G can be decomposed into ⌊ Δ ( G ) / 2 ⌋ cycles and a matching of size odd ( G ) / 2 . (ii) G can be decomposed into max ⁡ { odd ( G ) / 2 , ⌈ Δ ( G ) / 2 ⌉ } paths. (iii) G can be decomposed into ⌈ Δ ( G ) / 2 ⌉ linear forests. Each of these bounds is best possible. We actually derive (i)–(iii) from ‘quasirandom’ versions of our results. In that context, we also determine the edge chromatic number of a given dense quasirandom graph of even order. For all these results, our main tool is a result on Hamilton decompositions of robust expanders by Kühn and Osthus. [ABSTRACT FROM AUTHOR]

Our motivation is the question how similar thef-colouring problem is to the classic edge-colouring problem, particularly with regard to graph parameters. In 2010, Zhanget al.[On the fractional f-chromatic index of a graph, Int. J. Comput. Math. 87 (2010), pp. 3359–3369] gave a new description of thef-matching polytope and thereby derived a formula for the fractionalf-chromatic index stating that the fractionalf-chromatic index is equal to the maximum of the fractional maximumf-degree and the fractionalf-density. Unfortunately, this formula is incorrect. We present counterexamples for both the description of thef-matching polytope and the formula for the fractionalf-chromatic index. Finally, we prove a short lemma concerning the generalization of Goldberg's conjecture. [ABSTRACT FROM PUBLISHER]