On Minimum Sets of
1-Factors Covering a
Complete Multipartite
Graph
David Cariolaro1 and Hung-Lin Fu2 1INSTITUTE OF MATHEMATICS ACADEMIA SINICA, NANKANG, TAIPEI 11529, TAIWAN E-mail: cariolaro@math.sinica.edu.tw; davidcariolaro@hotmail.com 2DEPARTMENT OF APPLIED MATHEMATICS NATIONAL CHIAO TUNG UNIVERSITY HSIN CHU 30050 TAIWAN E-mail: hlfu@math.nctu.edu.tw
Received April 20, 2006; Revised January 21, 2008
Published online 15 April 2008 in Wiley InterScience(www.interscience.wiley.com). DOI 10.1002/jgt.20303
Abstract: We determine necessary and sufficient conditions for a com-plete multipartite graph to admit a set of 1-factors whose union is the whole graph and, when these conditions are satisfied, we determine the minimum size of such a set. © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 239–250, 2008
Keywords: 1–factor; 1–factor cover; excessive factorization; excessive index; complete multipartite graph
1. INTRODUCTION
All graphs considered will be finite, simple and undirected, unless stated otherwise. We denote byV (G) and E(G), respectively, the vertex and edge set of a graph G. Journal of Graph Theory
The order of G is|V (G)|. The maximum degree of G will be denoted by (G) and its chromatic index byχ(G).
Let G be a graph of even order. A 1-factor (or perfect matching) of G is a 1-regular spanning subgraph, that is, a set of exactly |V (G)|2 independent edges.1G
is 1-extendable if every edge of G belongs to at least one 1-factor of G. A 1-factor
cover of G is a setF of 1-factors of G such that ∪F∈F = E(G). Notice that G admits
a 1-factor cover if and only if it is 1-extendable. If G is 1-extendable, a 1-factor cover of minimum cardinality will be called an excessive factorization.
Thus, a 1-factorization of G is a 1-factor coverF with the property that all the 1-factors inF are pairwise disjoint. Any 1-factorization is an excessive factorization, but the converse is obviously not true. For example, the Petersen graph has no 1-factorization, but has an excessive factorization consisting of five 1-factors (see [2]). The graphs which admit an excessive factorization are precisely those that have a 1-factor cover, that is, those that are 1-extendable.
Let G be a 1-extendable graph. The excessive index of G, denoted χe(G), is the size of an excessive factorization of G. We define χe(G) = ∞ if G is not 1-extendable.
Bonisoli [1] and Wallis [6] considered 1-factor covers of the complete graphK2n
which do not contain a 1-factorization ofK2n.
Bonisoli and Cariolaro [2] introduced the concept of excessive factorization, defined the parameterχe(G), and studied excessive factorizations of regular graphs. They posed a number of open problems and conjectures. A first question is, of course, to determineχe(G) for any graph G. It is observed in [2] that this problem is NP-hard since, if G is regular and has even order, thenχe(G) = (G) if and only if G is 1-factorizable, and to determine whether a regular graph G is 1-factorizable is NP-complete. Therefore, we can expect to be able to determineχe(G) only for some specific classes of graphs.
In this article, we consider the class of complete multipartite graphs. Hoffman and Rodger [3] determined the chromatic index of all complete multipartite graphs. Here, we shall determine the excessive indexχe(G) of any complete multipartite graph G.
We will often use, without further reference, the following fact, proved by de Werra [7] and, independently, by McDiarmid [4]. If a multigraph G has a
k-edge coloring, that is, ifk ≥ χ(G), then it also has an equalized k-edge
color-ing, namely a k-edge coloring such that each color class has size either|E(G)|k or |E(G)|k .
We shall also need the concept of excessive coloring. An excessive coloring of a graph G is an assignment of (possibly more than one) colors to each of the edges of G such that the edges on which a given color appears are independent (i.e., they form a matching). Thus an excessive coloring can be simply specified
1To be precise, a 1-factor F of G is a 1-regular spanning subgraph of G and a perfect matching is the edge set of
as a collection of matchings of G whose union isE(G). It is normally interesting to consider this concept when additional restrictions are imposed on the matchings which form the color classes. For example, when each color class is a 1-factor, the corresponding excessive coloring is equivalent to a 1-factor cover.
2. 1-EXTENDABLE COMPLETE MULTIPARTITE GRAPHS
We adopt the notation G = K(n1, n2, . . . , nr) to designate a complete
multipar-tite graph with parmultipar-tite sets of sizen1, n2, . . . , nr, where n1≥ n2≥ n3≥ . . . ≥ nr. We also let V1, V2, . . . , Vr denote the r partite sets of G. By definition, for each
i, Vi is an independent set of ni vertices of G which are joined to every vertex inG − Vi.
Trivially, the complete bipartite graph K(m, n) is 1-extendable if and only if
m = n, in which case it actually has a 1-factorization. Therefore, χ
e(K(m, n)) = n
if n = m and ∞ otherwise. From now on we make the convention that all
com-plete multipartite graphs considered have r partite sets, where r ≥ 3. The fol-lowing lemma is probably well known, but we give a full proof for the sake of completeness.
Lemma 1. The graphG = K(n1, n2, . . . , nr) has a 1-factor if and only if
1. ri=1niis even;
2. n1≤
r i=2ni.
Proof. The first of the above conditions is clearly necessary in order for the graph G to have a 1-factor, as G must have even order. To see the necessity of the second, it suffices to see that any 1-factor of G must match the vertices ofV1
(the first partite set) to the vertices of the complement (since the vertices in V1
are mutually nonadjacent). Hence, the two conditions are necessary. To see the sufficiency, assume both conditions hold. We prove the existence of a 1-factor by induction on k, where 2k =ri=2ni− n1.
Ifk = 0, then we have n1=
r
i=2ni and a 1-factor of G is easily obtained by matching the vertices ofV1to the vertices ofV2∪ V3∪ · · · ∪ Vr. Assume now that the theorem holds for any G withri=2ni− n1< 2k and consider the case of a G
withri=2ni− n1= 2k. Let x ∈ Vrandy ∈ Vr−1and consider the edgee = xy. We
prove that there is a 1-factor of G containing this edge. This is equivalent to proving that the graphG − x − y has a 1-factor. But it is easily seen that the graph G − x − y is complete multipartite with partite sets V1, V2, . . . , Vr, where |Vi| = ni for all
i ≤ r − 2 and |V
r−1| = nr−1− 1 and |Vr| = nr− 1. Moreover, G − x − y satisfies the inductive hypothesis, since|V2| + |V3| + · · · + |Vr−1 | + |Vr| − |V1| = 2k − 2. Thus,G − x − y has a 1-factor and hence G has the desired 1-factor. Using Lemma 1, it is easy to determine which complete multipartite graphs admit an excessive factorization, as given by the following theorem.
Theorem 1. The graphK(n1, n2, . . . , nr) is 1-extendable if and only if
1. ri=1niis even;
2. n1<
r i=2ni.
Proof. Let G = K(n1, n2, . . . , nr) and suppose G is 1-extendable. The first
condition follows immediately from the fact that G has a 1-factor. Let e be an edge joining the second and third partite sets. By the fact that G is 1-extendable, there exists a 1-factor F containing e. Clearly, this 1-factor must match the vertices of the first partite set onto the vertices of the complement, but the two vertices which are the endpoint of e are F-saturated. Hence, the condition (2) above (which is clearly equivalent ton1≤
r
i=2ni− 2, given the parity of G) must hold. Conversely, suppose G satisfies both conditions above. We prove that G is 1-extendable. Let e ∈ E(G). We prove the existence of a 1-factor F containing e. Equivalently, we prove that the graphG − x − y has a 1-factor , where xy = e. With-out loss of generality, we can assumex ∈ Vi, y ∈ Vj, andi < j. But then G − x −
y ∼= G1= K(n1, n2, . . . ni−1, ni− 1, ni+1, . . . , nj−1, nj− 1, nj+1, . . . , nr) andG1
is easily seen to satisfy the hypotheses of Lemma 1. Hence,G1has a 1-factor and
G has the desired 1-factor.
3. SOME LEMMAS
Theorem 1 gives necessary and sufficient conditions for a complete multipartite graph G to admit an excessive factorization, that is, to satisfyχe(G) < ∞. We are now left with the task of determining preciselyχe(G) for all such graphs.
We start by giving some lower bounds. One obvious lower bound is the maximum degree of G, because every edge incident with a vertex of maximum degree must belong to a distinct 1-factor in an excessive factorization. Thus, χe(G) ≥ (G) holds, not only for complete multipartite graphs, but for all graphs G (in fact an easy argument along the same line also shows that, for all graphs G,χe(G) ≥ χ(G), but we shall not need this stronger inequality here).
The next lower bound is less trivial. LetG = K(n1, n2, . . . , nr). Let Vibe the
ith partite set of G. LetEibe defined as
Ei= E(G − Vi). Define σi(G) = 2|Ei| |V (G)| − 2|Vi| . (1)
Since the vertices ofViare independent in G, any 1-factor F of G must contain exactlyniedges joiningVitoG − Vi. Thus, in particular, the 1-factor F contains exactly |V (G)|−2|Vi|
contain at leastσi(G) 1-factors. This proves that
χ
e(G) ≥ max1≤i≤rσi(G).
As shown next, the quantity max1≤i≤rσi(G) is particularly simple to evaluate, since it is always equal toσ1(G).
Proposition 1. LetG = K(n1, n2, . . . , nr). Then
σ1(G) = max
1≤i≤rσi(G),
where the parametersσi(G) are defined in (1).
Proof. We start by observing that
σk(G) =
21≤i<j≤r;i,j =kninj r
i=1ni− 2nk .
Therefore, Proposition 1 follows from the truth of the following inequality in-volving positive integersx1, x2, . . . , xr, where x1= max1≤i≤rxi:
2≤i<j≤rxixj r i=1xi− 2x1 ≥ 1≤i<j≤r;i,j =kxixj r i=1xi− 2xk . (2)
By the arbitrariety of thexis (i > 1), we can assume k = 2. Further, if x1= x2
there is clearly nothing to prove because the two sides of (2) are in this case identical. Thus, we may assume thatx1> x2. We have to prove that
2≤i<j≤r xixj r i=1 xi− 2x2 ≥ 1≤i<j≤r;i,j =2 xixj r i=1 xi− 2x1 .
Letξ = x3+ x4+ · · · + xr and letη =
3≤i<j≤rxixj. Using these notations, we can rewrite the above inequality as
(x2ξ + η)(x1− x2+ ξ) ≥ (x1ξ + η)(x2− x1+ ξ).
Multiplying out, simplifying and rearranging the terms, we obtain (x1− x2)ξ2− (x12− x22)ξ − 2η(x1− x2)≤ 0.
Dividing byx1− x2, which is positive by assumption, we obtain
ξ2− (x 1+ x2)ξ − 2η ≤ 0. But we have ξ2= (x 3+ x4+ · · · + xr)2= x23+ x24+ · · · + x2r + 2 · 3≤i<j≤r xixj = x23+ x24+ · · · + x2r + 2η.
Thus, the problem is reduced to showing that x2 3+ x 2 4+ · · · + x 2 r ≤ (x1+ x2)(x3+ x4+ · · · + xr).
Using the assumption thatx1= max1≤i≤rxiand the fact that
x2
3+ x24+ · · · + x2r ≤ x1(x3+ x4+ · · · + xr)< (x1+ x2)(x3+ x4+ · · · + xr),
we conclude the proof.
Thus, we have two nontrivial lower bounds on χe(G), one is (G) and the other isσ1(G). Neither of the two is necessarily worse or better than the other. For
example, ifG = K(5, 4, 3) then σ1(G) = 12 > (G) = 9, but if G = K(4, 4, 4, 2)
thenσ1(G) = 11 < (G) = 12.
Let
τ(G) = max{σ1(G), (G)}.
By what we have just proved, we have
Lemma 2. Let G be a 1-extendable complete r-partite graph. Then χ
e(G) ≥ τ(G).
In the next section, we shall prove that the above inequality is indeed an equality by proving the following theorem.
Theorem 2. Let G be a 1-extendable complete r-partite graph. Then χe(G) = τ(G) = max{σ1(G), (G)}.
We shall use the following lemma, which can be seen as an extension of Hall’s Theorem. It generalizes a theorem of Bondy ([5, Theorem 13.3, p.109]). Inter alia, it completely solves the problem of characterizing the excessive colorings ofG − V1
which extend to 1-factor covers of G, for any complete multipartite graph G. Lemma 3. LetC be a set (of colors) and let s, t be positive integers, with s ≤ t ≤
|C|. Let T = {y1, y2, . . . , yt} be a set of cardinality t and let S = {x1, x2, . . . , xs}
be a set (disjoint from T) of cardinality s. For eachy ∈ T , let L(y) ⊂ C be a set
of colors. Let, for each α ∈ C, Tα= {y ∈ T ; | α ∈ L(y)}. Consider the complete
bipartite graph X with bipartition (T, S). There exists an excessive coloring ψ of X
with color setC such that, for each α ∈ C, the color class corresponding to α is a
perfect matching from S toTαif and only if the following conditions are satisfied:
1. everyα ∈ C is contained in precisely s sets of the family {L(y) | y ∈ T }; 2. |L(y)| ≥ s (for all y ∈ T ).
Proof. Assume that there exists an excessive coloring as in the statement of the lemma. Then clearly condition (1) is satisfied since the existence of a perfect matching from S toTα implies|Tα| = s. Condition (2) follows from the fact that
everyy ∈ T is incident with s edges in the graph X, and each of these edges must be assigned at least one distinct color byψ, this color being in L(y). Thus, the two conditions above are necessary for the existence ofψ. We now show that they are sufficient.
Consider the bipartite graphB1 with bipartition (T, C), where there is an edge
between y andα if and only if α ∈ L(y). By assumption, degB1(α) = s for all α ∈ C and degB
1(y) ≥ s for all y ∈ T .
LetB2be a spanning subgraph ofB1such that
degB2(y) = s for all y ∈ T.
SinceB2is bipartite and(B2)= s, by K˝onig’s Theorem B2has an s-edge coloring
π with colors {1, 2, . . . , s}. We now define an edge coloring θ of X as follows: if
y is joined inB2 to colorα by an edge colored j, we color the edge yxj of X by
colorα. It is easy to see that the coloring θ is well defined, and that it is in fact a proper edge coloring of X. To obtain the required excessive coloring of X, it is now sufficient, for each colorα, to extend arbitrarily the color class Cαcorresponding
toα to a perfect matching from S to Tα.
Instead of proving Theorem 2 directly, in the next section we shall prove the following theorem, whose equivalence with Theorem 2 will be established below. Theorem 3. Let G = K(n1, n2, . . . , nr) be a 1-extendable complete r-partite
graph and let H = K(n2, n3, . . . , nr). Then there exists an excessive coloringφ
of H with exactlyτ(G) colors such that each color class misses exactly n1vertices
of H and each vertex of H misses at leastn1colors.
Using Lemma 3, we now prove the equivalence between Theorems 2 and 3. Lemma 4. Theorem 2 holds for G if and only if Theorem 3 does.
Proof. Assume Theorem 2 is true for the graphG = K(n1, n2, . . . , nr). Letψ
be an excessive factorization of G. Thenψ, when viewed as an excessive coloring, consists ofτ(G) color classes. Let H = G − V1, whereV1is the largest partite set
of G. ThenH ∼= K(n2, n3, . . . , nr). Clearly, the restrictionφ of ψ to E(H) makes
Theorem 3 true for the graph G. Hence, if Theorem 2 holds for G then Theorem 3 does. For the converse, let G and H be as above and assume Theorem 3 holds for
G. By Theorem 3, there exists an excessive coloringφ of H with τ(G) color classes
such that each color class is a matching missing exactlyn1vertices of H and each
vertex of H misses at least n1colors. We now extendφ to an excessive coloring
of G as follows. LetC be the color set of φ. For each vertex v ∈ V (H), let L(v) be the set of colors missing atv, that is, the set of colors which do not appear on any of the edges incident withv. By the conditions satisfied by φ, |L(v)| ≥ n1 for all
v ∈ V (H) and every color α ∈ C appears on exactly n1of the setsL(v), v ∈ V (H).
Therefore, the conditions of Lemma 3 are satisfied by the color set C, the family of setsL = {L(v) | v ∈ V (H)} and the set S = V1. By Lemma 3, there exists an
such that, for every colorα, the color class corresponding to α is a perfect matching from S toTα, whereTαis the set of vertices inV (H) which are missing (with respect
toφ) color α. Therefore, it is easily seen that the map ρ defined by
ρ(e) =
φ(e) if e ∈ E(H), and ψ(e) if e ∈ E(U).
is an excessive coloring of G which usesτ(G) colors and such that every color class is a 1-factor of G. Therefore,ρ is a 1-factor cover of G consisting of τ(G) 1-factors, and, by Lemma 2, this number is necessarily the minimum, which proves thatρ is an excessive factorization of G. Thus, Theorem 2 holds for G. This concludes the
proof of Lemma 4.
4. THE MAIN RESULT
We now prove Theorem 2 for all those complete multipartite graphs for which
σ1(G) > (G) by proving the following.
Theorem 4. Let G be a 1-extendable complete multipartite graph such that σ1(G) > (G). Then χe(G) = σ1(G).
Proof. LetG = K(n1, n2, . . . , nr) and let H = G − V1∼= K(n2, n3, . . . , nr), where V1 is the largest partite set of G, and assume that G is 1-extendable and
σ1(G) > (G). By Lemma 4, it will suffice to show that Theorem 3 holds for G.
Thus, we need to find an excessive coloringφ of H with exactly σ1(G) color classes,
each of which misses exactlyn1vertices of H and with respect to which each vertex
of H misses at leastn1colors. Letm = |V (H)|−n2 1. Notice thatσ1(G) =
|E(H)| m
. Let
m1= |E(H)| − (σ1(G) − 1)m. Notice that 0 < m1≤ m.
We prove that there exists an edge-coloring of H withσ1(G) − 1 color classes of
size m and 1 color class of sizem1.
By assumption G has a 1-factor, and hence H has a matching of size m. LetM1
be a matching in H of sizem1. Consider the graphH − M1. We have
χ(H − M
1)≤ χ(H) ≤ χ(G) = (G) ≤ σ1(G) − 1.
Hence, there exists a (σ1(G) − 1)-edge coloring of H − M1. Notice that|E(H −
M1)| = (σ1(G) − 1)m. But then there exists an equalized (σ1(G) − 1)-edge
color-ing ofH − M1, so that each color class has size exactly m.
Putting back the matchingM1as an additional color class, we have the required
edge coloring of H. But now, in order to obtain an excessive coloring of H as in Theorem 3, we just need to extend the color classM1to an arbitrary color class
(matching) of size m of H. The excessive coloringφ thus defined is such that any vertexv of H of degree degH(v) “sees” at most degH(v) + 1 colors, because the only possible edges with multiple colors are those inM1. Thus, at any vertex of H
there are at most
colors missing. Notice that each color class ofφ is a matching of size m and hence (by the definition of m) misses exactlyn1vertices of H. Therefore,φ verifies Theorem 3
for G, and hence (by Lemma 4) it verifies Theorem 2 for G. This terminates the
proof.
To terminate the proof of Theorem 2, we need to settle the caseσ1(G) ≤ (G).
The following notation will be helpful in the sequel. If Z is a graph,v ∈ V (Z) and
k is an integer, the k-deficiency ofv in Z is the quantity
k def(v) = k − degZ(v)
and the k-deficiency of Z is defined as
k def(Z) =
v∈V (Z)
k def(v) =
v∈V (Z)
(k − degZ(v)).
Notice that, ifk = (G), then the k-deficiency of Z is usually called deficiency of
Z and denoted by def(Z).
Proposition 2. Let G = K(n1, n2, . . . , nr) and let H = G − V1∼=
K(n2, n3, . . . , nr). Then the following two conditions are equivalent:
1. σ1(G) ≤ (G);
2. (G) def(H) ≥ n1(G).
Proof. Condition 1 is equivalent to 2|E(H)| |V (H)| − n1 ≤ (G), that is, 2|E(H)| ≤ (|V (H)| − n1)(G), that is, v∈V (H) degH(v) ≤ (|V (H)| − n1)(G), that is, v∈V (H) (degH(v) − (G)) ≤ −n1(G).
Changing sign, we have
(G) def(H) ≥ n1(G)
which is condition 2.
We will need the following lemma.
Lemma 5. LetG = K(n1, n2, . . . , nr) be a 1-extendable complete multipartite
that there exists a multigraphH∗containing H as a spanning subgraph such thatH∗ is obtained by replicating some of the existing edges of H but without adding edges
between nonadjacent vertices of H. Suppose furthermore that(H∗)= (H) and
(G) def(H∗)≤ n
1(G). Then χe(G) = (G).
Proof. LetH, H∗be as in the statement of Lemma 5. The condition
(G) def(H∗)≤ n
1(G)
is equivalent (arguing as in Proposition 2) to |E(H∗)| ≥ 1
2(|V (H)| − n1)(G). (3) By possibly removing some of the edges fromH∗, we can assume that the sign of equality holds in (3) and hence that
|E(H∗)| = 1
2(|V (H)| − n1)(G), (4) which is equivalent to
(G) def(H∗)= n1(G). (5)
Letv ∈ V (H) = V (H∗). By assumption, the edges incident withv which are in
H∗but not in H are at most
(H) − degH(v) ≤ (H) − (|V (H)| − n1)= n1− nr ≤ n1− 1,
so that (denoting byµ(H∗) the maximum multiplicity of the edges ofH∗) we have
µ(H∗)≤ n
1, since H is a simple graph. But then, by Vizing’s Theorem, we have
χ(H∗)≤ (H∗)+ µ(H∗)≤ (H) + n
1= (G).
Thus,H∗ is(G)-edge colorable. But then, in particular, H∗has an equalized
(G)-edge coloring, which we denote by ϕ. It follows by (4) that each color class
ofϕ contains exactly 12(|V (H)| − n1) edges. Since(H∗)= (H) = (G) − n1,
it follows that every vertex ofH∗misses at leastn1of the colors given byϕ.
Letψ be the excessive coloring of H obtained by assigning to the edge xy ∈ E(H)
all the colors assigned by ϕ to the edges xy ∈ E(H∗). Then clearlyψ is an ex-cessive coloring of H using (G) colors, such that each color class contains exactly 12(|V (H)| − n1) edges and each vertex misses at least n1 colors. Thus,
ψ satisfies Theorem 3 and hence, by Lemma 4, Theorem 2 holds for G, as
we wanted.
We are ready to prove the following theorem, which, together with Theorem 4, proves Theorem 2.
Theorem 5. Let G be a 1-extendable complete multipartite graph such that σ1(G) ≤ (G). Then χe(G) = (G).
Proof. By Lemma 5, it suffices to prove the existence of a multigraphH∗ as specified in the statement of Lemma 5.
LetH∗be a maximal multigraph which is a spanning supergraph of H obtained by replicating existing edges of H without introducing edges between nonadja-cent vertices of H and such that (H∗)= (H). We prove that H∗ satisfies the conditions of Lemma 5.
Claim 1. There can be at most one partite set Vi ofH∗ containing vertices of
degree less than(H).
This is obvious since otherwise we could add toH∗ an edge by replicating an existent edgexy of H∗without violating the constraints on the maximum degree but contradicting the maximality ofH∗.
Conclusion. If all the vertices inV (H∗) have degree(H) there is clearly nothing to prove, since then
(G) def(H∗)= n
1|V (H)| ≤ n1(G)
and all the conditions of Lemma 5 are satisfied.
Thus, we can assume (by Claim 1) that there is exactly one partite setViofH∗ containing vertices of degree less than(H).
But then (H) def(H∗)= v∈Vi ((H) − degH∗(v)) ≤ v∈Vi ((H) − degH(v)) = ni(ni− nr). Hence (G) def(H∗)≤ n i(ni− nr)+ n1(|V (G)| − n1). (6)
Using the fact thatn1≥ ni≥ nr, it is easily seen that
ni(ni− nr)≤ n1(n1− nr).
Hence, using (6), we see that
(G) def(H∗)≤ n
1(n1− nr)+ n1(|V (G)| − n1)= n1(|V (G)| − nr)= n1(G).
Therefore, H∗ satisfies all the conditions of Lemma 5 and hence Theorem 5 is
proved.
ACKNOWLEDGMENT
The authors are grateful to the anonymous referees for their comments on the first version of this article.
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