Open Problem Garden

¿Are critical k-forests tight? ★★

Author(s): Strausz

Conjecture

Let $H$ be a $k$ -uniform hypergraph. If $H$ is a critical $k$ -forest, then it is a $k$ -tree.

Keywords: heterochromatic number

Posted by Dino
updated May 5th, 2014

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Bigger cycles in cubic graphs ★★

Author(s):

Problem Let $G$ be a cyclically 4-edge-connected cubic graph and let $C$ be a cycle of $G$ . Must there exist a cycle $C' \neq C$ so that $V(C) \subseteq V(C')$ ?

Keywords: cubic; cycle

Posted by mdevos
updated September 4th, 2009

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Graph Theory » Basic G.T. » Matchings

The intersection of two perfect matchings ★★

Author(s): Macajova; Skoviera

Conjecture Every bridgeless cubic graph has two perfect matchings $M_1$ , $M_2$ so that $M_1 \cap M_2$ does not contain an odd edge-cut.

Keywords: cubic; nowhere-zero flow; perfect matching

Posted by mdevos
updated August 30th, 2007

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Reloid corresponding to funcoid is between outward and inward reloid ★★

Author(s): Porton

Conjecture For any funcoid $f$ and reloid $g$ having the same source and destination $( \mathsf{\tmop{RLD}})_{\tmop{out}} f \subseteq g \subseteq ( \mathsf{\tmop{RLD}})_{\tmop{in}} f \Leftrightarrow ( \mathsf{\tmop{FCD}}) g = f.$

Keywords: funcoid; inward reloid; outward reloid; reloid

Posted by porton
updated September 2nd, 2007

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Topology

Distributivity of union of funcoids corresponding to reloids ★★

Author(s): Porton

Conjecture $\bigcup \left\langle ( \mathsf{\tmop{FCD}}) \right\rangle S = ( \mathsf{\tmop{FCD}}) \bigcup S$ if $S\in\mathscr{P}\mathsf{RLD}(A;B)$ is a set of reloids from a set $A$ to a set $B$ .