One-adhesive polymatroids

Adhesive polymatroids were defined by F. Mat\'u\v{s} motivated by entropy functions. Two polymatroids are adhesive if they can be glued together along their joint part in a modular way; and are one-adhesive, if one of them has a single point outside their intersection. It is shown that two polymatroids are one-adhesive if and only if two closely related polymatroids have any extension. Using this result, adhesive polymatroid pairs on a five-element set are characterized

A short proof of the G\'acs--K\"orner theorem

We present a short proof of a celebrated result of G\'acs and K\"orner giving sufficient and necessary condition on the joint distribution of two discrete random variables $X$ and $Y$ for the case when their mutual information matches the extractable (in the limit) common information. Our proof is based on the observation that the mere existence of certain random variables jointly distributed with $X$ and $Y$ can impose restriction on all random variables jointly distributed with $X$ and $Y$

Connected Graph Game

Two players claim alternately edges of the complete graph on n vertices. The winner is the one who makes the spanned subgraph with the claimed edges connected. We determine, for each n, who wins the game. The avoidance version is also considered, where the player who is forced to make the spanned subgraph to be connected, loses the game