7
votes
2answers
131 views

Bounds on chromatic number of $k$-planar graphs

A $1$-planar graph can be drawn in the plane so that each arc is crossed at most once by another arc. A $k$-planar graph can be drawn so that each arc is crossed at most $k$ times. Planar graphs are ...
5
votes
1answer
290 views

coloring in lattice

This is a mathematical question raised from engineering and physics: Is there some established mathematical approach in filling a physical lattice with some colored basis (black and white here)? For ...
5
votes
2answers
250 views

Fractional chromatic number, find reference to a particular alternate definition for

I'm searching for a reference to a particular alternate definition of the fractional chromatic number of graphs. Let me review the most common definition and basic properties first. Let $ G $ be ...
1
vote
1answer
405 views

Regular graph colorings

[Since I didn't get any feedback at MSE, I dare to post this question here, too.] Call a coloring $C:V(G) \rightarrow \lbrace 1,\dots,n \rbrace$ of a graph $G$ regular when every vertex with color ...
5
votes
2answers
507 views

4-coloring maps of pentagons

Is there a simple proof for the 4-color theorem when restricted to (finite) maps all of whose regions are pentagons? I am in fact most interested in convex pentagons, if that additional structure ...