AbstractThe first part of the paper studies star-cycle factors of graphs. It characterizes star-cycle factors of a graph G and proves upper bounds for the minimum number of $$K_{1,2}$$
K
1
,
2
-components in a $$\{K_{1,1}, K_{1,2}, C_n:n\ge 3\}$$
{
K
1
,
1
,
K
1
,
2
,
C
n
:
n
≥
3
}
-factor of a graph G. Furthermore, it shows where these components are located with respect to the Gallai–Edmonds decomposition of G and it characterizes the edges which are not contained in any $$\{K_{1,1}, K_{1,2}, C_n:n\ge 3\}$$
{
K
1
,
1
,
K
1
,
2
,
C
n
:
n
≥
3
}
-factor of G. The second part of the paper proves that every edge-chromatic critical graph G has a $$\{K_{1,1}, K_{1,2}, C_n:n\ge 3\}$$
{
K
1
,
1
,
K
1
,
2
,
C
n
:
n
≥
3
}
-factor, and the number of $$K_{1,2}$$
K
1
,
2
-components is bounded in terms of its fractional matching number. Furthermore, it shows that for every edge e of G, there is a $$\{K_{1,1}, K_{1,2}, C_n:n\ge 3\}$$
{
K
1
,
1
,
K
1
,
2
,
C
n
:
n
≥
3
}
-factor F with $$e \in E(F)$$
e
∈
E
(
F
)
. Consequences of these results for Vizing’s critical graph conjectures are discussed.