Forcing a 2-regular minor ★★

Author(s): Reed; Wood

Conjecture   Every graph with average degree at least $ \frac{4}{3}t-2 $ contains every 2-regular graph on $ t $ vertices as a minor.

Keywords: minors

Fractional Hadwiger ★★

Author(s): Harvey; Reed; Seymour; Wood

Conjecture   For every graph $ G $,
(a) $ \chi_f(G)\leq\text{had}(G) $
(b) $ \chi(G)\leq\text{had}_f(G) $
(c) $ \chi_f(G)\leq\text{had}_f(G) $.

Keywords: fractional coloring, minors

Direct proof of a theorem about compact funcoids ★★

Author(s): Porton

Conjecture   Let $ f $ is a $ T_1 $-separable (the same as $ T_2 $ for symmetric transitive) compact funcoid and $ g $ is a uniform space (reflexive, symmetric, and transitive endoreloid) such that $ ( \mathsf{\tmop{FCD}}) g = f $. Then $ g = \langle f \times f \rangle^{\ast} \Delta $.

The main purpose here is to find a direct proof of this conjecture. It seems that this conjecture can be derived from the well known theorem about existence of exactly one uniformity on a compact set. But that would be what I call an indirect proof, we need a direct proof instead.

The direct proof may be constructed by correcting all errors an omissions in this draft article.

Direct proof could be better because with it we would get a little more general statement like this:

Conjecture   Let $ f $ be a $ T_1 $-separable compact reflexive symmetric funcoid and $ g $ be a reloid such that
    \item $ ( \mathsf{\tmop{FCD}}) g = f $; \item $ g \circ g^{- 1} \sqsubseteq g $.

Then $ g = \langle f \times f \rangle^{\ast} \Delta $.

Keywords: compact space; compact topology; funcoid; reloid; uniform space; uniformity