Decompositions into spanning rainbow structures

Abstract

A subgraph of an edge‐coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs goes back more than 200 years to the work of Euler on Latin squares and has been the focus of extensive research ever since. Euler posed a problem equivalent to finding properly n‐edge‐coloured complete bipartite graphs Kn,n which can be decomposed into rainbow perfect matchings. While there are proper edge‐colourings of Kn,n without even a single rainbow perfect matching, the theme of this paper is to show that with some very weak additional constraints one can find many disjoint rainbow perfect matchings. In particular, we prove that if some fraction of the colour classes have at most (1−o(1))n edges, then one can nearly‐decompose the edges of Kn,n into edge‐disjoint perfect rainbow matchings. As an application of this, we establish in a very strong form a conjecture of Akbari and Alipour and asymptotically prove a conjecture of Barat and Nagy. Both these conjectures concern rainbow perfect matchings in edge‐colourings of Kn,n with quadratically many colours. The above result also has implications to some conjectures of Snevily about subsquares of multiplication tables of groups. Finally, using our techniques, we also prove a number of results on near‐decompositions of graphs into other rainbow structures like Hamiltonian cycles and spanning trees. Most notably, we prove that any properly coloured complete graph can be nearly‐decomposed into spanning rainbow trees. This asymptotically proves the Brualdi–Hollingsworth and Kaneko–Kano–Suzuki conjectures which predict that a perfect decomposition should exist under the same assumptions.

A counterexample to Stein’s Equi-$n$-square Conjecture

“…Again, one can ask an easier question by adding an extra condition forbidding symbol repetitions in rows and columns. Such a result is true and will be proved in [13]:…”

Decompositions into isomorphic rainbow spanning trees

“…Moreover, since {i} × D i,1 ∈ F, at least (1 − γ 515 )|D i,1 | elements of {i} × D i,1 are covered by M, which means that for at least (1 − γ 515 )|D i,1 | colours c in D i,1 , we have i ∈ I c and therefore c ∈ D ′ i . Thus, |D ′ i | ≥ (1−γ 515 )|D i,1 | ≥ (1−2γ)qn by (21). Hence, |D ′ i | = (1±2γ)qn, as required.…”

“…Thus, D ′ i ⊆ D i . In particular, we have |D ′ i | ≤ |D i | ≤ (1 + 2γ)qn by (21). Moreover, since {i} × D i,1 ∈ F, at least (1 − γ 515 )|D i,1 | elements of {i} × D i,1 are covered by M, which means that for at least (1 − γ 515 )|D i,1 | colours c in D i,1 , we have i ∈ I c and therefore c ∈ D ′ i .…”

Halfway to Rota’s Basis Conjecture

“…The closest one to Rota's basis conjecture seems to be the Brualdi-Hollingsworth conjecture [6], which posits that for every (n − 1)-edge-colouring of the complete graph K n , the edges can be decomposed into rainbow spanning trees. This conjecture has recently seen some exciting progress (see for example [22,27,3,25]). We wonder if some of the ideas developed for the study of rainbow structures could be profitably applied to Rota's basis conjecture.…”