Fractional matchings and component-factors of (edge-chromatic critical) graphs

The paper studies component-factors of graphs which can be characterized in terms of their fractional matching number. These results are used to prove that every edge-chromatic critical graph has a $[1,2]$-factor. Furthermore, fractional matchings of edge-chromatic critical graphs are studied and some questions are related to Vizing's conjectures on the independence number and 2-factors of edge-chromatic critical graphs.


Introduction and Motivation
We consider finite simple graphs. For a graph G, V (G) and E(G) denote the set of vertices and the set of edges, respectively. For a vertex v of V (G), E G (v) denotes the set of edges which are incident to v. The degree of v, denoted by d G (v), is |E G (v)|. The maximum degree of a vertex of G is denoted by ∆(G) and the minimum degree of a vertex of G is denoted by δ(G). If ∆(G) = δ(G) = k, then G is k-regular. If G is a 2-regular graph then it is also called a cycle, and if G is a connected 2-regular graph, then we also call G a circuit. For v ∈ V (G), the set of neighbors of v is denoted by N G (v). Clearly, d G (v) = |E G (v)| = |N G (v)|, for graphs. For a set X ⊆ V (G), the neighborhood of X is defined as N G (X) = x∈X N G (x).
For S ⊆ V (G), the set of edges with precisely one end in S is denoted by ∂ G (S). For A, B ⊂ V (G), the set of edges with one end in A and the other in B is denoted by E G (A, B).
Hence, E G (S, V (G) − S) = ∂ G (S). If there is no harm of confusion, then we will omit the indices.
A set M (M ⊆ E(G) or M ⊂ V (G)) is independent, if no two elements of M are adjacent.
An independent set of edges is also called a matching of G. The maximum cardinality of a matching of G is the matching number of G, which is denoted by µ(G). A matching M with The following theorems characterize graphs which satisfy relaxed conditions. These results had been generalized by Berge and Las Vergnas [4] to star-cycle factors. For each finite graph G there is an integer n such that iso(G − S) ≤ n|S| for all S ⊆ V (G).
Consequently, the following statement is proved. Corollary 1.7. 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 K 1,1 .
In section 3 we study edge-chromatic critical graphs. The edge-chromatic number χ ′ (G) of a graph G is the minimum number k of matchings which are needed to cover the edge set of G. In 1965, Vizing [18] proved that χ ′ (G) ∈ {∆(G), ∆(G) + 1} for a graph G. For k ≥ 2, a graph G is k-critical, if ∆(G) = k, χ ′ (G) = k + 1 and χ ′ (H) ≤ k for each proper subgraph H of G. We often say that G is a critical graph, if there is a k, such that G is a k-critical graph.
The maximum cardinality of an independent set of vertices is the independence number of G which is denoted by α(G). The following two conjectures are due to Vizing. Conjecture 1.8 ( [19]). If G is a critical graph, then G has a 2-factor.
Clearly, if Conjecture 1.8 is true, then Conjecture 1.9 is also true. Conjectures on factors on critical graphs are surveyed in [3] where it was conjectured that every critical graph has a [1,2] factor. We will prove this conjecture in section 3.
The article closes with section 4, where we study fractional matchings on critical graphs.

Fractional matching number and star-cycle factors
A graph G is factor-critical if G−v has a perfect matching for each v ∈ V (G). Analogously, a matching is near perfect if it covers all vertices but one. Let D(G) be the set of vertices of G which are missed by at least one maximum matching of G, let A(G) = N (D(G))−D(G) and . We call the triple (D(G), A(G), C(G)) a Gallai-Edmonds decomposition of G. If there is no harm of confusion we shortly write (D, A, C) instead of (D(G), A(G), C(G)). We will use the fundamental Gallai-Edmonds structure theorem.  M is a maximum matching of G}. 11]). Let G be a graph and n ≥ 0 be an integer. If µ f (G) = 1 2 (|V (G)| − n), then n = def (G) − nc(G).
Let G be a graph with µ f (G) = 1 2 (|V (G)| − n). Scheinerman [14] (Theorem 2.2.6) proved that n = max{iso(G − S) − |S| : S ⊆ V (G)}. We call a set S with iso(G − S) = |S| + n a witness for µ f (G). A crucial point in the proof of Theorem 2.2 is that every non-trivial component of G[D] has a fractional perfect matching. The following theorem shows that they have even more structural properties. A maximum fractional matching f with M ⊆ supp(f ) is a canonical maximum fractional matching of G (with respect to M ).
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 A(G) contains a witness for µ f (G). We will state this fact in a more detailed manner in the following corollary. If F is a star-cycle factor of G, then t F i denotes the number of K 1,i -components of F and let l( F is a star-cycle factor of G}. The next theorem gives a detailed insight into the structure of graphs with respect to their fractional matching number. Theorem 2.5. Let G be a graph, n ≥ 0 be an integer and λ be the minimum integer such that iso(G − S) ≤ λ|S| for all S ⊆ V (G). If µ f (G) = 1 2 (|V (G)| − n), then λ ≤ ⌈ n δ(G) ⌉ + 1 and G has a {K 1,1 , . . . , K 1,λ , C m : m ≥ 3}-factor F , such that l(G) = λ i=1 (i − 1)t F i = n. Furthermore, the K 1,j -components are induced subgraphs of G, and for j ≥ 2, their center We construct a sequence of subgraphs F 0 , . . . , F n of G, where the subgraph F i is the desired and the statement follows with Theorem 1.2, that is, t F i = 0 for each i ≥ 2 and therefore, l(G) = 0 and t 0 = 1 = λ.
Suppose that F k has been constructed in G k for k, with k ≤ n − 1. We will construct F k+1 Hence, there is a a ′ ∈ T A with d F k (a ′ ) < λ. Let p = d k+1 , a 1 , d 1 , . . . , a t , d t , a ′ be a minimal F k -alternating path (d i ∈ D(G) and a i ∈ A(G)) with end vertices d k+1 and a ′ . Note 2 , if e is an edge of a circuit of F , and f ′ (e) = 0 otherwise, is a fractional matching of G and e∈E(G) f ′ (e) = 1 2 (|V (G)| − n). It remains to show that λ ≤ ⌈ n δ(G) ⌉ + 1. Without loss of generality we may assume that Then in the above construction t ≤ λ and t increases at most by 1 all δ(G) steps. Hence, Since λ is minimum, the statement follows. circuits.
Corollary 2.7. Let G be a graph that has a {K 1,1 , . . . , K 1,λ , C m : m ≥ 3}-factor. Then Proof. By Corollary 2.6 G has a star-cycle factor with nc(G) odd cycles and l(G) vertices Theorem 2.8. Let G be a graph and e ′ ∈ E(G). If there is a maximum fractional matching for all e ∈ E(G) and f ′ (e ′ ) = 0, and the components of supp(f ′ ) are K 1,1 's or odd circuits.
Proof. Let f be a maximum fractional matching and e ′ ∈ E(G) with f (e ′ ) = 0. By Theorem for an integer n ≥ 0. Let f 0 be a maximum fractional matching with f 0 (e ′ ) = 0 and |{e : e ∈ E(G) and f 0 (e) = 0}| maximal, and let H = G[supp(f 0 )]. We will prove the statement by induction on n. n = 0: In this case, f and f 0 are fractional perfect matchings of G, and our proof of the statements closely follows the line of the proof of Theorem 1.1 given in [14].
If H contains an edge e 0 = vw with d H (v) = 1, then f 0 (e 0 ) = 1 and e 0 is the edge of a

Claim 1. H does not contain an even circuit.
Suppose to the contrary that it contains an even circuit C.
which assigns 0 to at least one more edge than f 0 , a contradiction.
if e ∈ E(G) and for the edges If n = 1, then h is a fractional perfect matching of G x and therefore, it is maximum.
For n ≥ 2 we suppose to the contrary that the graph G x has a fractional matching h 0 , a contradiction and the claim is proved.
for all e ∈ E(G) is the desired maximum fractional matching of G. The other direction of the statement is trivial.
Let min(G, K 1,2 ) = min{t F 2 : F is a {K 1,1 , K 1,2 , C m : m ≥ 3}-factor of G}. The following corollary will be used in section 3. Proof. The result follows directly from Theorem 2.5 and Corollary 2.6.
Theorem 1.2 is the special case m = n of the following corollary.
Corollary 2.11. Let G be a graph and let n, m be integers with 0 < n ≤ m ≤ 2n. If Proof. (i) Since 1 ≤ m n ≤ 2 it follows with Theorem 1.4 that G has a {K 1,1 , K 1,2 , C m : m ≥ 3}-factor. Furthermore, for all S ⊆ V (G): Now, the result follows with Corollaries 2.6 and 2.10.
(ii) By (i), G has as a {K 1,1 , K 1,2 , C m : m ≥ 3}-factor F with min(G, In the following we will apply Lovász' (g, f )-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 ( [12]). Let G be a multigraph and let g, f : V (G) → Z be functions such Then G has a (g, f )-factor if and only if for all disjoint subsets S and T of V (G), Notice that q ⋆ (S, T ) = 0 for all disjoint subsets S and Furthermore, the inequalities of (ii) are tight.
Proof. The condition iso(G−S) ≤ 2|S| in (ii) is satisfied, since G has a {K 1,1 , K 1,2 , C m : m ≥ 3}-factor. Therefore, it remains to prove that 2|S| − 2 ≤ iso(G − S). We first consider the graph G ′ which is obtained from G by contracting e, that is V ( If G has a {K 1,1 , K 1,2 , C m : m ≥ 3}-factor F with e ∈ F and e is contained in a C mcomponent, then decompose this component into K 1,1 and K 1,2 -components. So e is either contained in a K 1,1 -component or in a K 1,2 -component. Contract e, and the remaining edges factor of G and in any case, e is an end edge of a path. If we decompose all paths of length at least three into paths of length one or two, then we get a {K 1,1 , K 1,2 , C m : m ≥ 3}-factor F of G with e ∈ F , and the claim is proved. "⇐ ": Let S be a set of V (G) with u, v ∈ S and 2|S| − 2 ≤ iso(G − S). Let S ′ be the corresponding set of S. Since u, v ∈ S, we have w ∈ S ′ . Further |S| = |S ′ | + 1, By Theorem 2.12, G ′ has no (g ′ , f ′ )-factor and by Claim 1 G has no {K 1,1 , K 1,2 , C m : m ≥ 3}factor that contains e. Since G has a {K 1,1 , K 1,2 , C m : m ≥ 3}-factor, G also has a (g, f )-factor with g(x) = 1 and f (x) = 2 for all x ∈ V (G) and by Theorem 2.12 for all disjoint subsets X and Y of V (G) we have Since e is not contained in all {K 1,1 , K 1,2 , C m : m ≥ 3}-factors of G, by Claim 1 and Theorem 2.12, there exits two disjoint subsets X ′ and to g ′ and f ′ ). Let S ′ and T ′ be two subsets of V (G ′ ) satisfying γ(S ′ , T ′ ) < 0.
Case 2: w ∈ T ′ . We have again a contradiction.
Case 3: w ∈ S ′ . We have

Component factors of edge-chromatic critical graphs
Woodall [21] proved that α(G) ≤ 3 5 |V (G)| for a critical graph G. Using his proof approach we generalize some of his results to deduce that every critical graph has a [1, 2]-factor.
Clearly, every [1, 2]-factor can be decomposed into a {K 1,1 , K 1,2 , C m : m ≥ 3}-factor. We will use this fact to prove an upper bound for min(G, K 1,2 ) for critical graphs. Let G be a critical graph. We denote by σ(v, w) the number of vertices in N (w) \ {v} that have degree at least 2∆(G) − d G (v) − d G (w) + 2, for an edge vw of G. We have since by Lemma 3.1, w has at least ∆(G) − d G (v) + 1 neighbors different from v with degree ∆(G). 20]). Let G be a critical graph and v ∈ V (G) and let Then v has at least Theorem 3.3. Let G be a critical graph and let S be an arbitrary subset of V (G). Then Proof. Let G be a critical graph, S be an arbitrary subset of V (G) and T = Iso(G − S).
In a critical graph there are no vertices of degree less We define two functions f i : The functions g 1 and g 2 are both decreasing functions of k.
Let t be a vertex of T + and k := d G (t). Then , then the fraction is nonnegative. Thus the claim is proved.
We now define three charge functions M i , i ∈ {1, 2, 3} on V (G) as follows: We prove that the functions M 1 and M 2 satisfy This implies and therefore, |S|.
(i) Starting with the distribution M 0 , let each vertex in T receive charge 2 from each of its neighbors in S. Let the resulting charge distribution be called , with strict inequality if s is a vertex of S with fewer than ∆(G) neighbors in T . There exists such a vertex s, since either s has a neighbor in V (G) \ (S ∪ T ) or S ∪ T = V (G) and S is not an independent set. Thus, (ii) Starting with the distribution M 1 , we will redistribute charge according to the following discharging rule: the actual discharging rule than it would receive under the equitable discharging rule.
We fix a vertex t ∈ T − and denote by k the degree of t, so k = d G (t). Further we define a function h with h : N × N 0 → R and We prove, that vertex t gets at least 2(∆(G) − k) charge in Step 2. This implies that . We define p as in (4) of Lemma 3.2. It follows, that t has at least k − p − 1 neighbors s ∈ S with σ(s, t) ≥ ∆(G) − p − 1. Let N + (t) be a set of such k − (p + 1) neighbors and let N − (t) = N (t) \ N + (t). The set N − (t) contains p + 1 neighbors s of t, for each with σ(s, t) ≥ ∆(G) − k + p + 1, by the definition of p. Applying Claim 2 to the vertices N − (t) with l = p for the vertices in N − (t) and l = k − p − 2 for the vertices in N + (t), we see that t receives charge of at least M + (k, p) in Step 2, where It remains to show that the minimal value of M + is at least 2(∆(G)−k). Let r = p+1, (3) The derivative of this with respect to r is This is zero if and only if r = 1 2 k (unless ak − bk 2 = 0, if M + (k, p) is independent of p); thus, M + (k, p), regarded as a function of p, has only one stationary point (for positive p), when p + 1 = 1 2 k. Substituting this value of p gives where the inequality holds, because k < 1 2 ∆(G) and so To complete the proof, we must consider also the other extreme value of p, p = 0, and show that M + (k, 0) ≥ 2(∆(G) − k), so we have to show that This evidently holds with equality if k = 2; so we may assume that k ≥ 3. Since k < 1 2 ∆(G), we can write ∆(G) = 2k + q, where q ≥ 1. Ignoring the first term of (5), and dividing through by k − 1 and rearranging, it suffices to show that Since the left side of (6) is clearly an increasing function of q, it suffices to verify inequality (6) for s = 1, when the left side becomes which is positive since k ≥ 3.
and the proof is complete.   Proof. Let G be a critical graph and let e = vw. Suppose to the contrary that there is no {K 1,1 , K 1,2 , C m : m ≥ 3}-factor that contains e. By Theorems 3.3 and 2.13 there exists a subset S of V (G) with u, v ∈ S and 2|S| − 2 ≤ iso(G − S) < 3 2 − 1 ∆(G) |S| < 3 2 |S|. Since u, v ∈ S, |S| ≥ 2.

Fractional matchings on edge-chromatic critical graphs
The study of fractional matchings of critical graphs gives insight into the structure of critical graphs. Our studies of component factors of critical graphs uses the concept of fractional matchings. We propose the following conjecture.   In [9] it is proved, that G is critical if and only if G ′ is critical. Similar to the proofs of the corresponding statements for Conjectures 1.8 and 1.9 [3,15] we can apply Meredith extension to prove the following statement.
Theorem 4.3. The following two statements are equivalent for each k ≥ 3: matching. In [9] it is shown that for all k ≥ 3 there are k-critical graphs of even order which have no 1-factor, and that there are k-critical graphs G of odd order and G − v does not have a 1-factor, where d G (v) = δ(G). We close with the conjecture which is unsolved even for critical graphs which have a near perfect matching. However, it is true if Conjecture 4.1 is true.
Conjecture 4.4. Let k ≥ 3 and G be a k-critical graph. If G does not have a 1-factor, then µ f (G) > µ(G).