5

Part of 2015 239 Open Mathematical Olympiad

Problems(2)

Number of All Even Coordinates Points on a Polyhedron

Source: 239 2015 S P5

The nodes of a three dimensional unit cube lattice with all three coordinates even are coloured red and blue otherwise. A convex polyhedron with all vertices red is given. Assuming the number of red points on its border is

n

. How many blue vertices can be on its border?

geometry3D geometry

another chromatic problem from 239

Source: 239 2015 J5

Edges of a complete graph with

2m

vertices are properly colored with

2m-1

colors. It turned out that for any two colors all the edges colored in one of these two colors can be described as union of several

4

-cycles. Prove that

m

2

graph theorycombinatoricsedge coloring