Mathematics > Combinatorics
[Submitted on 22 Aug 2015 (v1), last revised 21 Sep 2016 (this version, v2)]
Title:A note on the Erdös-Faber-Lovász Conjecture: quasigroups and complete digraphs
View PDFAbstract:A decomposition of a simple graph $G$ is a pair $(G,P)$ where $P$ is a set of subgraphs of $G$, which partitions the edges of $G$ in the sense that every edge of $G$ belongs to exactly one subgraph in $P$. If the elements of $P$ are induced subgraphs then the decomposition is denoted by $[G,P]$.
A $k$-$P$-coloring of a decomposition $(G,P)$ is a surjective function that assigns to the edges of $G$ a color from a $k$-set of colors, such that all edges of $H\in P$ have the same color, and, if $H_1,H_2\in P$ with $V(H_1)\cap V(H_2)\neq\emptyset$ then $E(H_1)$ and $E(H_2)$ have different colors. The \emph{chromatic index} $\chi'((G,P))$ of a decomposition $(G,P)$ is the smallest number $k$ for which there exists a $k$-$P$-coloring of $(G,P)$.
The well-known Erdös-Faber-Lovász Conjecture states that any decomposition $[K_n,P]$ satisfies $\chi'([K_n,P])\leq n$. We use quasigroups and complete digraphs to give a new family of decompositions that satisfy the conjecture.
Submission history
From: Christian Rubio-Montiel [view email][v1] Sat, 22 Aug 2015 17:29:05 UTC (6 KB)
[v2] Wed, 21 Sep 2016 08:37:31 UTC (40 KB)
References & Citations
Bibliographic and Citation Tools
Bibliographic Explorer (What is the Explorer?)
Connected Papers (What is Connected Papers?)
Litmaps (What is Litmaps?)
scite Smart Citations (What are Smart Citations?)
Code, Data and Media Associated with this Article
alphaXiv (What is alphaXiv?)
CatalyzeX Code Finder for Papers (What is CatalyzeX?)
DagsHub (What is DagsHub?)
Gotit.pub (What is GotitPub?)
Hugging Face (What is Huggingface?)
Papers with Code (What is Papers with Code?)
ScienceCast (What is ScienceCast?)
Demos
Recommenders and Search Tools
Influence Flower (What are Influence Flowers?)
CORE Recommender (What is CORE?)
arXivLabs: experimental projects with community collaborators
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.