# Recent Activity

## Giuga's Conjecture on Primality ★★

Author(s): Giuseppe Giuga

\begin{conjecture} $p$ is a prime iff $~\displaystyle \sum_{i=1}^{p-1} i^{p-1} \equiv -1 \pmod p$ \end{conjecture}

Keywords: primality

## Extremal problem on the number of tree endomorphism ★★

Author(s): Zhicong Lin

\begin{conjecture} An endomorphism of a graph is a mapping on the vertex set of the graph which preserves edges. Among all the $n$ vertices' trees, the star with $n$ vertices has the most endomorphisms, while the path with $n$ vertices has the least endomorphisms. \end{conjecture}

Keywords:

## Coloring the Odd Distance Graph ★★★

Author(s): Rosenfeld

The \emph{Odd Distance Graph}, denoted ${\mathcal O}$, is the graph with vertex set ${\mathbb R}^2$ and two points adjacent if the distance between them is an odd integer.

\begin{question} Is $\chi({\mathcal O}) = \infty$? \end{question}

Keywords: coloring; geometric graph; odd distance

## Cores of Cayley graphs ★★

Author(s): Samal

\begin{conjecture} Let $M$ be an abelian group. Is the \Def[core]{core (graph theory)} of a \Def{Cayley graph} (on some power of $M$) a Cayley graph (on some power of $M$)? \end{conjecture}

Keywords: Cayley graph; core

## Triangle free strongly regular graphs ★★★

Author(s):

\begin{problem} Is there an eighth triangle free strongly regular graph? \end{problem}

Keywords: strongly regular; triangle free

## Graph product of multifuncoids ★★

Author(s): Porton

\begin{conjecture} Let $F$ is a family of multifuncoids such that each $F_i$ is of the form $\lambda j \in N \left( i \right) : \mathfrak{F} \left( U_j \right)$ where $N \left( i \right)$ is an index set for every $i$ and $U_j$ is a set for every $j$. Let every $F_i = E^{\ast} f_i$ for some multifuncoid $f_i$ of the form $\lambda j \in N \left( i \right) : \mathfrak{P} \left( U_j \right)$ regarding the filtrator $\left( \prod_{j \in N \left( i \right)} \mathfrak{F} \left( U_j \right) ; \prod_{j \in N \left( i \right)} \mathfrak{P} \left( U_j \right) \right)$. Let $H$ is a graph-composition of $F$ (regarding some partition $G$ and external set $Z$). Then there exist a multifuncoid $h$ of the form $\lambda j \in Z : \mathfrak{P} \left( U_j \right)$ such that $H = E^{\ast} h$ regarding the filtrator $\left( \prod_{j \in Z} \mathfrak{F} \left( U_j \right) ; \prod_{j \in Z} \mathfrak{P} \left( U_j \right) \right)$. \end{conjecture}

Keywords: graph-product; multifuncoid

## Atomicity of the poset of multifuncoids ★★

Author(s): Porton

\begin{conjecture} The poset of multifuncoids of the form $(\mathscr{P}\mho)^n$ is for every sets $\mho$ and $n$: \begin{enumerate} \item atomic; \item atomistic. \end{enumerate} \end{conjecture}

See below for definition of all concepts and symbols used to in this conjecture.

Refer to \href[this Web site]{http://www.mathematics21.org/algebraic-general-topology.html} for the theory which I now attempt to generalize.

Keywords: multifuncoid

## Atomicity of the poset of completary multifuncoids ★★

Author(s): Porton

\begin{conjecture} The poset of completary multifuncoids of the form $(\mathscr{P}\mho)^n$ is for every sets $\mho$ and $n$: \begin{enumerate} \item atomic; \item atomistic. \end{enumerate} \end{conjecture}

See below for definition of all concepts and symbols used to in this conjecture.

Refer to \href[this Web site]{http://www.mathematics21.org/algebraic-general-topology.html} for the theory which I now attempt to generalize.

Keywords: multifuncoid

## Cycle double cover conjecture ★★★★

Author(s): Seymour; Szekeres

\begin{conjecture} For every graph with no \Def[bridge]{bridge (graph theory)}, there is a list of cycles so that every edge is contained in exactly two. \end{conjecture}

Keywords: cover; cycle

## Upgrading a completary multifuncoid ★★

Author(s): Porton

Let $\mho$ be a set, $\mathfrak{F}$ be the set of filters on $\mho$ ordered reverse to set-theoretic inclusion, $\mathfrak{P}$ be the set of principal filters on $\mho$, let $n$ be an index set. Consider the filtrator $\left( \mathfrak{F}^n ; \mathfrak{P}^n \right)$.

\begin{conjecture} If $f$ is a completary multifuncoid of the form $\mathfrak{P}^n$, then $E^{\ast} f$ is a completary multifuncoid of the form $\mathfrak{F}^n$. \end{conjecture}

See below for definition of all concepts and symbols used to in this conjecture.

Refer to \href[this Web site]{http://www.mathematics21.org/algebraic-general-topology.html} for the theory which I now attempt to generalize.

Keywords:

## 4-regular 4-chromatic graphs of high girth ★★

Author(s): Grunbaum

\begin{problem} Do there exist 4-regular 4-chromatic graphs of arbitrarily high girth? \end{problem}

Keywords: coloring; girth

## Perfect cuboid ★★

Author(s):

\begin{conjecture} Does a perfect cuboid exist? % Enter your conjecture in LaTeX % You may change "conjecture" to "question" or "problem" if you prefer. \end{conjecture}

Keywords:

## Forcing a $K_6$-minor ★★

Author(s): Barát ; Joret; Wood

\begin{conjecture} Every graph with minimum degree at least 7 contains a $K_6$-minor. \end{conjecture}

\begin{conjecture} Every 7-connected graph contains a $K_6$-minor. \end{conjecture}

Keywords: connectivity; graph minors

## Funcoidal products inside an inward reloid ★★

Author(s): Porton

\begin{conjecture} (solved) If $a \times^{\mathsf{\ensuremath{\operatorname{RLD}}}} b \subseteq \left( \mathsf{\ensuremath{\operatorname{RLD}}} \right)_{\ensuremath{\operatorname{in}}} f$ then $a \times^{\mathsf{\ensuremath{\operatorname{FCD}}}} b \subseteq f$ for every funcoid $f$ and atomic f.o. $a$ and $b$ on the source and destination of $f$ correspondingly. \end{conjecture}

A stronger conjecture:

\begin{conjecture} If $\mathcal{A} \times^{\mathsf{\ensuremath{\operatorname{RLD}}}} \mathcal{B} \subseteq \left( \mathsf{\ensuremath{\operatorname{RLD}}} \right)_{\ensuremath{\operatorname{in}}} f$ then $\mathcal{A} \times^{\mathsf{\ensuremath{\operatorname{FCD}}}} \mathcal{B} \subseteq f$ for every funcoid $f$ and $\mathcal{A} \in \mathfrak{F} \left( \ensuremath{\operatorname{Src}}f \right)$, $\mathcal{B} \in \mathfrak{F} \left( \ensuremath{\operatorname{Dst}}f \right)$. \end{conjecture}

Keywords: inward reloid

## Odd cycles and low oddness ★★

Author(s):

\begin{conjecture} If in a bridgeless cubic graph $G$ the cycles of any $2$-factor are odd, then $\omega(G)\leq 2$, where $\omega(G)$ denotes the oddness of the graph $G$, that is, the minimum number of odd cycles in a $2$-factor of $G$. \end{conjecture}

Keywords:

## Odd perfect numbers ★★★

Author(s): Ancient/folklore

\begin{conjecture} There is no odd \Def{perfect number}. \end{conjecture}

Keywords: perfect number

## Matching cut and girth ★★

Author(s):

\begin{question} For every $d$ does there exists a $g$ such that every graph with average degree smaller than $d$ and girth at least $g$ has a matching-cut? \end{question}

Keywords: matching cut, matching, cut

## Strong 5-cycle double cover conjecture ★★★

Author(s): Arthur; Hoffmann-Ostenhof

\begin{conjecture} Let $C$ be a circuit in a bridgeless cubic graph $G$. Then there is a five cycle double cover of $G$ such that $C$ is a subgraph of one of these five cycles. \end{conjecture}

Keywords: cycle cover

## 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

## Characterizing (aleph_0,aleph_1)-graphs ★★★

Call a graph an $(\aleph_0,\aleph_1)$-\emph{graph} if it has a bipartition $(A,B)$ so that every vertex in $A$ has degree $\aleph_0$ and every vertex in $B$ has degree $\aleph_1$.
\begin{problem} Characterize the $(\aleph_0,\aleph_1)$-graphs. \end{problem}