# Rainbow Odd Cycles

@article{Aharoni2020RainbowOC,
title={Rainbow Odd Cycles},
author={Ron Aharoni and Joseph Briggs and Ron Holzman and Zilin Jiang},
journal={SIAM J. Discret. Math.},
year={2020},
volume={35},
pages={2293-2303}
}
• Published 19 July 2020
• Mathematics
• SIAM J. Discret. Math.
We prove that every family of (not necessarily distinct) odd cycles $O_1, \dots, O_{2\lceil n/2 \rceil-1}$ in the complete graph $K_n$ on $n$ vertices has a rainbow odd cycle (that is, a set of edges from distinct $O_i$'s, forming an odd cycle). As part of the proof, we characterize those families of $n$ odd cycles in $K_{n+1}$ that do not have any rainbow odd cycle. We also characterize those families of $n$ cycles in $K_{n+1}$, as well as those of $n$ edge-disjoint nonempty subgraphs of $K_{n… ## Figures from this paper • Mathematics • 2019 For two graphs$G$and$H$, write$G \stackrel{\mathrm{rbw}}{\longrightarrow} H$if$G$has the property that every {\sl proper} colouring of its edges admits a {\sl rainbow} copy of$H$. We study • Mathematics Order • 2022 This is a survey paper on rainbow sets (another name for “choice functions”). The main theme is the distinction between two types of choice functions: those having a large (in the sense of belonging • Mathematics • 2022 We prove that every family of (not necessarily distinct) even cycles D 1 , . . . , D ⌊ 6( n − 1) 5 ⌋ +1 on some ﬁxed n -vertex set has a rainbow even cycle (that is, a set of edges from distinct D i We study the following problem. How many distinct copies of H can an n -vertex graph G have, if G does not contain a rainbow F , that is, a copy of F where each edge is contained in a diﬀerent copy • Mathematics • 2022 We prove that a family T of distinct triangles on n given vertices that does not have a rainbow triangle (that is, three edges, each taken from a diﬀerent triangle in T , that form together a ## References SHOWING 1-10 OF 15 REFERENCES • Mathematics Combinatorica • 2019 It is proved that any family of (not necessarily distinct) sets of edges in an$r$-uniform hypergraph, each having a fractional matching of size$n$has a rainbow fractional matches of size £n. • Mathematics • 2019 For a given class$\mathcal{C}$of graphs and given integers$m \leq n$, let$f_\mathcal{C}(n,m)$be the minimal number$k$such that every$k$independent$n$-sets in any graph belonging to • Mathematics Israel Journal of Mathematics • 2022 Let G = ( V, E ) be a graph and n a positive integer. Let I n ( G ) be the abstract simplicial complex whose simplices are the subsets of V that do not contain an independent set of size n in G . We • Mathematics Electron. J. Comb. • 2009 The conjecture that equality holds for all values of$r,s and $t$ is proved and the conjecture is a strengthening of a famous conjecture, described below, of Ryser, Brualdi and Stein.
It is shown that anm×nrow-latin rectangle with symbols in {1,2,?,k},k?n, has a transversal wheneverm?2n?1, and that this lower bound formis sharp. Several applications are given. One is the
• Mathematics
ArXiv
• 2020
This work shows that if a binary matroid of rank r is colored with exactly r colors, then M either contains a rainbow colored circuit or a monochromatic cut, and gives a complete characterization of minimally rigid graphs admitting such a coloring.
• Materials Science
Graphs Comb.
• 2019
It is shown that whenever the edges of a connected simple graph on n vertices are colored with n-1 colors appearing so that no cycle in G is rainbow, there must be a monochromatic edge cut in G, and that $$n-1$$ is the largest integer for which the main theorem holds.