1 Introduction and Motivation
We consider finite simple graphs. For a graph , and denote the set of vertices and the set of edges, respectively. For a vertex of , denotes the set of edges which are incident to . The degree of , denoted by , is . The maximum degree of a vertex of is denoted by and the minimum degree of a vertex of is denoted by . If , then is -regular. If is a 2-regular graph then it is also called a cycle, and if is a connected 2-regular graph, then we also call a circuit. For , the set of neighbors of is denoted by . Clearly, , for graphs. For a set , the neighborhood of is defined as . For , the set of edges with precisely one end in is denoted by . For , the set of edges with one end in and the other in is denoted by . Hence, . If there is no harm of confusion, then we will omit the indices.
A set ( or ) is independent, if no two elements of are adjacent. An independent set of edges is also called a matching of . The maximum cardinality of a matching of is the matching number of , which is denoted by . A matching with is a maximum matching of . The number of vertices which are not incident to an edge of a maximum matching is the matching-deficiency of , and it is denoted by . Clearly, .
A fractional matching of is a function such that for all . If for each edge, then
is the characteristic function of a matching of. The fractional matching number is . Clearly, and if , then is a fractional perfect matching. For a fractional matching the set is the support of and it is denoted by .
Theorem 1.1 ( (Theorem 2.1.5)).
For any graph , is an integer. Moreover, there is a fractional matching for which and for every .
Let be a graph and be two functions such that for all . A -factor is a spanning subgraph of that satisfies . If and for all , then is a -factor, and if , then is a -factor of . Clearly, if is a 1-factor, then is a perfect matching of . If is a factor of a graph , then a path is -alternating, if its edges are in and alternately.
For a set of connected graphs, a spanning subgraph of is called an -factor if each component of is isomorphic to an element of . If , then a component of which is isomorphic to is called an -component of . The number of components of is denoted by . A component is trivial if it consists of a single vertex and non-trivial otherwise. The set of trivial components of is denoted by and denotes .
The complete bipartite graph with bipartition and , is denoted by . In case of , is called a star and the vertex of degree is its center vertex. For , either of the two vertices can be regarded as its center vertex. A -factor of is called a star-cycle factor.
For a set of vertices let and be the subgraph of induced by and , respectively. The following theorems characterize some component factors of graphs.
Theorem 1.2 ().
A graph has a -factor if and only if for all .
In terms of fractional perfect matchings, Theorem 1.2 is equivalent to the following formulation.
Theorem 1.3 ().
A graph has a fractional perfect matching if and only if for all .
The following theorems characterize graphs which satisfy relaxed conditions.
Theorem 1.4 ().
A graph has a -factor if and only if for all .
These results had been generalized by Berge and Las Vergnas  to star-cycle factors.
Theorem 1.6 ().
Let be a graph and be a function, and let . The graph has a star-cycle factor such that
if is the center vertex of a star component of , and
for each circuit component of
if and only if .
For each finite graph there is an integer such that for all . Consequently, the following statement is proved.
Every graph has a star-cycle factor.
In section 2 we characterize graphs with specific star-cycle factors in terms of their fractional matching number. In particular, we give an upper bound for the size of a star and for the number of star components which are different from .
In section 3 we study edge-chromatic critical graphs. The edge-chromatic number of a graph is the minimum number of matchings which are needed to cover the edge set of . In 1965, Vizing  proved that for a graph . For , a graph is -critical, if , and for each proper subgraph of . We often say that is a critical graph, if there is a , such that is a -critical graph. The maximum cardinality of an independent set of vertices is the independence number of which is denoted by . The following two conjectures are due to Vizing.
Conjecture 1.8 ().
If is a critical graph, then has a 2-factor.
Conjecture 1.9 ().
If is a critical graph, then .
Clearly, if Conjecture 1.8 is true, then Conjecture 1.9 is also true. Conjectures on factors on critical graphs are surveyed in  where it was conjectured that every critical graph has a factor. We will prove this conjecture in section 3.
The article closes with section 4, where we study fractional matchings on critical graphs.
2 Fractional matching number and star-cycle factors
A graph is factor-critical if has a perfect matching for each . Analogously, a matching is near perfect if it covers all vertices but one. Let be the set of vertices of which are missed by at least one maximum matching of , let and . We call the triple a Gallai-Edmonds decomposition of . If there is no harm of confusion we shortly write instead of . We will use the fundamental Gallai-Edmonds structure theorem.
Let be a graph. If is a Gallai-Edmonds decomposition of , then
every component of is factor-critical,
has a perfect matching,
every maximum matching consists of a near perfect matching on each component of , a perfect matching on , and a matching which matches every vertex of to one distinct component of , and
Next we formulate a sharpening of this result in the context of fractional matchings. Let be a maximum matching of a graph and be the number of non-trivial components of that are not matched by an edge , and .
Theorem 2.2 ().
Let be a graph and be an integer. If , then .
Let be a graph with . Scheinerman  (Theorem 2.2.6) proved that . We call a set with a witness for . A crucial point in the proof of Theorem 2.2 is that every non-trivial component of has a fractional perfect matching. The following theorem shows that they have even more structural properties.
Theorem 2.3 ().
Let be a factor-critical graph with . Then has a fractional perfect matching with for every and the set forms exactly one odd circuit.
forms exactly one odd circuit.
Furthermore, every maximum matching of is contained in the support of a fractional matching with values in . Let be a maximum matching with . A maximum fractional matching with is a canonical maximum fractional matching of (with respect to ).
Theorem 2.2 shows that every graph has a canonical maximum fractional matching. A look into the proof details of Theorem 2.2 yields that it is also shown that contains a witness for . We will state this fact in a more detailed manner in the following corollary.
Let be a graph, be an integer, and . If is a canonical maximum fractional matching w.r.t. , then contains two disjoint subsets and with
induces a perfect matching on ; in particular, , and
is a witness for .
If is a star-cycle factor of , then denotes the number of -components of and let . The next theorem gives a detailed insight into the structure of graphs with respect to their fractional matching number.
Let be a graph, be an integer and be the minimum integer such that for all . If , then and has a -factor , such that . Furthermore, the -components are induced subgraphs of , and for , their center vertices are in and their leaves are in .
Let be a canonical maximum fractional matching w.r.t. . For we have and for let . Let , and for let and . Clearly, is a subgraph of and is a canonical maximum fractional matching of w.r.t. .
We construct a sequence of subgraphs of , where the subgraph is the desired -factor on () and .
If , then has a perfect fractional matching, for all and the statement follows with Theorem 1.2, that is, for each and therefore, and .
Suppose that has been constructed in for , with . We will construct in .
Case A: There is a vertex with or . Then is a -factor of . The factor is obtained from by extending a -component to a -component. Hence, and . Furthermore, . Thus, .
Case B: For all : . Let be the set of all vertices of and for which there is an -alternating path with initial vertex , and . Note, that , since is a canonical maximum fractional matching w.r.t. .
If for all , then, by the definition of and , it follows that is a set of isolated vertices in . But , a contradiction to the choice of .
Hence, there is a with . Let be a minimal -alternating path ( and ) with end vertices and . Note that , , and . Let be obtained from by interchanging the edges of and in . Hence, is a -factor of . As in Case A it follows that and .
Let . Then is a -factor of and . We cannot do better since with if is an edge of a -component of , , if is an edge of a circuit of , and otherwise, is a fractional matching of and .
It remains to show that . Without loss of generality we may assume that . Let be the constructed -factor. Then in the above construction and increases at most by 1 all steps. Hence, . Therefore, for all . Since is minimum, the statement follows. ∎
For each graph : and has a -factor with circuits.
Let be a graph that has a -factor. Then
By Corollary 2.6 has a star-cycle factor with odd cycles and vertices extend -components to a -components, . Therefore, . ∎
Let be a graph and . If there is a maximum fractional matching of with , then there is a maximum fractional matching with for all and , and the components of are ’s or odd circuits.
Let be a maximum fractional matching and with . By Theorem 1.1 we have that for an integer . Let be a maximum fractional matching with and maximal, and let . We will prove the statement by induction on .
If contains an edge with , then and is the edge of a -component of . Hence, for all . In particular, .
does not contain an even circuit.
Suppose to the contrary that it contains an even circuit . Let and if , then let . Let . Define , with if and for let and . Then is a maximum fractional matching with and which assigns 0 to at least one more edge than , a contradiction.
If contains an odd circuit , then is a circuit component of .
Suppose that contains a vertex with . Let be a path which starts in with an edge which is not an edge of . This path cannot return to , since then would contain an even circuit. It can also not have an end vertex of degree 1, since then for the edge which is incident to in . Hence, it ends at a vertex with . Thus, contains a graph which consists of two odd circuits and which are connected by a path (possibly of length 0). Let be a function with if and alternately on the path which connects the two odd circuits of and alternately around the circuits such that for each . If , then choose such that . Let be the smallest number such that there is an edge with . Then is fractional perfect matching of which assigns the value 0 to more edges that . Furthermore, the value 0 can only achieved on an edge with . Hence, and we obtain a contradiction to the definition of . Thus, the claim is proved.
Hence, the components of are odd circuits or ’s. The function with , if is an edge of a circuit component of , , if is an edge of a component of and , if is the desired fractional perfect matching of with .
: For let . Let be the set of vertices of with . Add a vertex and edges for to to obtain a new graph . Note that .
Extend to a function with if and for the edges , choose appropriately such that and . The function is a fractional matching on . It holds that .
is a maximum fractional matching of .
If , then is a fractional perfect matching of and therefore, it is maximum.
For we suppose to the contrary that the graph has a fractional matching with and . It follows that , a contradiction and the claim is proved.
By definition, and therefore, is a maximum fractional matching on with and .
By induction hypothesis, there is a maximum fractional matching of with for all and . Since it follows that . Suppose to the contrary that is a vertex of a circuit component of . Since is an odd circuit, has a perfect matching. Thus, , a contradiction. Hence, is a vertex of a -component of , and with for all is the desired maximum fractional matching of . ∎
Let be a graph and . There is a maximum fractional matching of with if and only if is an edge of a maximum cycle-star factor of .
() Let for an integer . By Theorem 2.8 there is a fractional maximum
with for all and . Hence, is an edge of
a circuit or a -component of . Furthermore, there are precisely vertices with
. Let . Then . If
is a vertex of a circuit component of , then, since is of odd order, we easily deduce a
contradiction to the maximality of . Hence, is a vertex of a -component of .
Furthermore, at most one endvertex of a -component can be in , since for otherwise we again can deduce a contradiction to the maximality of .
Extending by connecting each to one of its neighbors yields the
desired -factor of .
The other direction of the statement is trivial. ∎
If , with minimal, then the cycle-star factor in Corollary 2.9 is not necessarily a -factor with .
Let . The following corollary will be used in section 3.
Let be a graph, that has a -factor and let be a natural number. Then, if and only if .
Theorem 1.2 is the special case of the following corollary.
Let be a graph and let be integers with . If for all subsets , then
Since it follows with Theorem 1.4 that has a -factor. Furthermore, for all :
Since for all it follows that
By , has as a -factor with . Then, for all we have
In the following we will apply Lovász’ -factor Theorem. This the the only theorem, where multigraphs are allowed. Here a multigraph is a graph that may have loops and multiple edges.
Theorem 2.12 ().
Let be a multigraph and let be functions such that for all . Then has a -factor if and only if for all disjoint subsets and of ,