We don't really need to reduce to the cubic case.

as a control mechanism, they store the minds of their (former) enemies consciously immersed in simulations of their (respective) reality(ies). i had the rare experience of accidentally melding with such a mind---the Mind-Time channel's rotating modulation code glitched or something. the reality was really real, pain and all that, just minor suggestive control over the avatar in space.   the rewards were enough to want to exert any control i could...

protesting the brantification of "landon"

my graph software
here counting even/odd spanning eulerian subgraphs

defense

unwinding

trees

The structure of paradox.

How to win at solitaire when a demon is mucking with the cards.

A slightly simpler proof of Kostochka's lemma. It avoids the need for one extra lemma.

For a collection of maximum cliques Q in a graph G, let X_Q be the
intersection graph of Q.

Lemma. If Q is a collection of maximum cliques in a graph G with
\omega(G) > 2/3 (\Delta(G) + 1) such that X_Q is connected, then \cap
Q \neq \emptyset.

Proof. Suppose not and choose a counterexample Q := {Q_1, ..., Q_r}
minimizing r.

Let A be a noncutvertex in X_Q and B a neighbor of A. Put Z := Q -
{A}. Then X_Z is connected and hence by minimality of r, \cap Z \neq
\emptyset. In particular, |\cup Z| \leq \Delta(G) + 1.

Hence |\cup Q| \leq |\cup Z| + |A - B| \leq 2(\Delta(G) + 1) -
\omega(G) < 2\omega(G). This contradicts Hajnal's lemma.

Conjecture 14 from my recent paper is false by some pretty easy examples.


However, i still think Conjecture 13 (which Conjecture 14 implies) is true.