Distance graphs with maximum chromatic number

The distance graph G(D)G(D) has the set of integers as vertices and two vertices are adjacent in G(D)G(D) if their difference is contained in the set D⊂ZD⊂Z. A conjecture of Zhu states that if the chromatic number of G(D)G(D) achieves its maximum value |D|+1|D|+1 then the graph has a triangle. The conjecture is proven to be true if |D|⩽3|D|⩽3. We prove that the chromatic number of a distance graph with D={a,b,c,d}D={a,b,c,d} is five only if either D={1,2,3,4k}D={1,2,3,4k} or D={a,b,a+b,b-a}D={a,b,a+b,b-a}. This confirms a stronger version of Zhu's conjecture for |D|=4|D|=4, namely, if the chromatic number achieves its maximum value then the graph contains K4K4.