One-factorizations of the complete graph Kp+1K_{p+1} arising from parabolas

There are three types of affine regular polygons in AG(2, q): ellipse, hyperbola and parabola. The first two cases have been investigated in previous papers. In this note, a particular class of geometric one-factorizations of the complete graph Kn arising from parabolas is constructed and described in full detail. With the support of computer aided investigation, it is also conjectured that up to isomorphisms this is the only one-factorization where each one-factor is either represented by a line or a parabola

Coloring decompositions of complete geometric graphs

A decomposition of a non-empty simple graph $G$ is a pair $[G,P]$, such that $P$ is a set of non-empty induced subgraphs of $G$, and every edge of $G$ belongs to exactly one subgraph in $P$. The chromatic index $\chi'([G,P])$ of a decomposition $[G,P]$ is the smallest number $k$ for which there exists a $k$-coloring of the elements of $P$ in such a way that: for every element of $P$ all of its edges have the same color, and if two members of $P$ share at least one vertex, then they have different colors. A long standing conjecture of Erd\H{o}s-Faber-Lov\'asz states that every decomposition $[K_n,P]$ of the complete graph $K_n$ satisfies $\chi'([K_n,P])\leq n$. In this paper we work with geometric graphs, and inspired by this formulation of the conjecture, we introduce the concept of chromatic index of a decomposition of the complete geometric graph. We present bounds for the chromatic index of several types of decompositions when the vertices of the graph are in general position. We also consider the particular case in which the vertices are in convex position and present bounds for the chromatic index of a few types of decompositions.Comment: 18 pages, 5 figure

Factorizations of complete multigraphs

In this paper, several general results are obtained on the Oberwolfach problem that provide isomorphic 2-factorizations of 2K. One consequence of these results is that the existence of a 2-factorization in which each 2-factor of 2K consists of one cycle of length x and one of length n-x is completely settled. The techniques used to obtain these results are novel, using for example the Lindner-Rodger generalizations of Marshall Hall's classic embedding theorem for incomplete latin squares

11-factorizations of complete multigraphs arising from finite geometry

Some interesting classes of $1$-factorizations and $1$-hyperfactorizations of the complete graph $K_{2n}$ are related to ovals in finite projective planes. We construct a new infinite family of $1$-factorizations of complete multigraphs arising from some combinatorial properties of ovals and unitals in projective planes over finite fields

One-factorizations of complete multigraphs arising from maximal (k;n)(k;n)-arcs in PG(2,2h)\mathrm{PG}(2,2^{h})

A new class of one-factorizations of the complete multigraph $\lambda K_{2^{h+1}+2}$ is constructed for every $\lambda =2^{k}$, with $1\leq k\leq h$. Furthermore, some connections with blocking sets in $\mathrm{PG}(2,2^{h})$ are established