SEMIREGULAR FACTORIZATIONS OF REGULAR MULTIGRAPHS

An (r,r+1)-factor of a graph G is a spanning subgraph H such that dH(v)∈{r,r+1} for all vertices v∈ (G). If G is expressed as the union of edge-disjoint (r,r+1)-factors, then this expression is an (r,r+1)-factorization of G. Let μ(r) be the smallest integer with the property that if G is a regular loopless multigraph of degree d with d≥μ(r), then G has an (r,r+1)-factorization. It is shown that $\mu (r)\le \frac {3}{2}r^2+3r+1$ if r is even. The proof employs a novel list-coloring approach. Together with known results, this shows that μ(r)=r2+1 if r is odd and $\frac {3}{2}(r^2-2r)\le \mu (r)\le \min \{2r^2-3r, \frac {3}{2}r^2+3r+1\}$ if r is even.