Rainbow Odd Cycles

  title={Rainbow Odd Cycles},
  author={Ron Aharoni and Joseph Briggs and Ron Holzman and Zilin Jiang},
  journal={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

Anti-Ramsey properties of randomly perturbed dense graphs

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

Choice Functions

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

Rainbow even cycles

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

Rainbow copies of $F$ in families of $H$

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

Rainbow triangles in families of triangles

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



Rainbow Fractional Matchings

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.

Rainbow independent sets in certain classes of graphs.

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

Complexes of graphs with bounded independence number

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

Rainbow Matchings in r-Partite r-Graphs

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.

Transversals in Row-Latin Rectangles

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

A generalization of carathéodory's theorem

Rainbow and monochromatic circuits and cuts in binary matroids

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.

On Rainbow-Cycle-Forbidding Edge Colorings of Finite Graphs

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.