# Jaeger, Francois

## Mapping planar graphs to odd cycles ★★★

Author(s): Jaeger

\begin{conjecture} Every planar graph of girth $\ge 4k$ has a homomorphism to $C_{2k+1}$. \end{conjecture}

Keywords: girth; homomorphism; planar graph

## The additive basis conjecture ★★★

Author(s): Jaeger; Linial; Payan; Tarsi

\begin{conjecture} For every prime $p$, there is a constant $c(p)$ (possibly $c(p)=p$) so that the union (as multisets) of any $c(p)$ bases of the vector space $({\mathbb Z}_p)^n$ contains an additive basis. \end{conjecture}

Keywords: additive basis; matrix

## A nowhere-zero point in a linear mapping ★★★

Author(s): Jaeger

\begin{conjecture} If ${\mathbb F}$ is a finite field with at least 4 elements and $A$ is an invertible $n \times n$ matrix with entries in ${\mathbb F}$, then there are column vectors $x,y \in {\mathbb F}^n$ which have no coordinates equal to zero such that $Ax=y$. \end{conjecture}

Keywords: invertible; nowhere-zero flow

## Petersen coloring conjecture ★★★

Author(s): Jaeger

\begin{conjecture} Let $G$ be a \Def[cubic]{cubic graph} graph with no \Def[bridge]{bridge (graph theory)}. Then there is a coloring of the edges of $G$ using the edges of the \Def[Petersen]{petersen graph} graph so that any three mutually adjacent edges of $G$ map to three mutually adjancent edges in the Petersen graph. \end{conjecture}

Keywords: cubic; edge-coloring; Petersen graph

## Jaeger's modular orientation conjecture ★★★

Author(s): Jaeger

\begin{conjecture} Every $4k$-\Def[edge-connected]{connectivity (graph theory)} graph can be oriented so that ${\mathit indegree}(v) - {\mathit outdegree}(v) \cong 0$ (mod $2k+1$) for every vertex $v$. \end{conjecture}

Keywords: nowhere-zero flow; orientation