Graphs of exact colorings ★★

Author(s):

Conjecture For $  c \geq m \geq 1  $, let $  P(c,m)  $ be the statement that given any exact $  c  $-coloring of the edges of a complete countably infinite graph (that is, a coloring with $  c  $ colors all of which must be used at least once), there exists an exactly $  m  $-colored countably infinite complete subgraph. Then $  P(c,m)  $ is true if and only if $  m=1  $, $  m=2  $, or $  c=m  $.

Keywords:

Imbalance conjecture ★★

Author(s): Kozerenko

Conjecture   Suppose that for all edges $ e\in E(G) $ we have $ imb(e)>0 $. Then $ M_{G} $ is graphic.

Keywords: edge imbalance; graphic sequences