Albinia consists of two states: Alexton and Brighton. Each road in Albinia connects two towns from Alexton and Brighton respectively. It is known that no town is connected with more than 10 others.
Prove that it is possible to color all roads in Albinia, using 10 colors, in such a way that no two adjacent roads would be the same color (we call two roads adjacent if they leave the same town).
When there are 10 towns in each of states A and B, with each A-town Ai connected to each B-town Bj by a road of color c, i,j,c=0,...,9, the conditions of the problem can be satisfied by choosing c=c(i,j)=i+j mod 10. When the states have no more than 10 towns, we can, as necessary, just delete towns from the 10 x 10 case. When a state has more than 10 towns, perhaps we can again fall back on the 10 x 10 case by choosing 10 primary towns and allocating the roads of the nonprimary towns to suitable primary towns.
Posted by Richard
on 2004-03-05 13:26:18