Covering Graphs: The Covering Problem Solved

Abstract

For every fixed graphH, we determine theH-covering number ofK_n, for alln>n₀(H). We prove that ifhis the number of edges ofH, andgcd(H)=dis the greatest common divisor of the degrees ofH, then there existsn₀=n₀(H), such that for alln>n₀,$\text{C(H,}\text{K}{}_{\mathrm{n}}\text{)=}\text{dn}\text{2h}\text{n−1}\text{d}\text{,}$unlessdis even,n=1 mod dandn(n−1)/d+1=0 mod(2h/d), in which case[formula]Our main tool in proving this result is the deep decomposition result of Gustavsson.