Extensions of SomeFactorization Results fromSimple Graphs toMultigraphs

S. I. El-ZanatiM. J. Plantholt

S. K. TipnisDEPARTMENT OF MATHEMATICS

ILLINOIS STATE UNIVERSITYNORMAL, ILLINOIS 61790-4520

E-mail: mikep@math.ilstu.edu

ABSTRACT

It was shown in a recent paper that an rs-regular multigraphGwith maximum multiplicityµ(G) ≤ r can be factored into r regular simple graphs if first we allow the deletion of arelatively small number of hamilton cycles from G. In this paper, we use this theoremto obtain extensions of some factorization results on simple graphs to new results onmultigraphs. c© 1997 John Wiley & Sons, Inc.

1. INTRODUCTION

For a multigraph G, we denote by V (G), E(G), and ∆(G) its vertex set, edge set and maximumdegree, and by µ(G) its multiplicity, the maximum number of parallel edges joining any pairof vertices in G. For undefined terminology and notation see [1]. Throughout this paper, afactorization of G will mean a partition of E(G) and if H is a subgraph of G, then G −H willdenote the subgraph of G with vertex set V (G) and edge set E(G)−E(H). In [4] the followingmultigraph result was proved.

Theorem 1 [4]. Let G be an rs-regular multigraph of order n with µ(G) ≤ r, where n and r areeven. If s ≥ n/2 then G contains r/2 hamilton cycles C1, C2, . . . , Cr/2 such that E(G− C1 −

Journal of Graph Theory Vol. 24, No. 4, 291 295 (1997)c© 1997 John Wiley & Sons, Inc. CCC 0364-9024/97/040291-05

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C2 − · · · − Cr/2) = E(H1) ∪ E(H2) · · · ∪ E(Hr), where H1, H2, . . . , Hr are (s − 1)-regularedge-disjoint simple graphs.

A similar result was obtained in [4] for the cases when n is odd and s < n/2. In someinstances, the result in Theorem 1 makes it possible to obtain multigraph factorization resultsfrom factorization results on simple graphs. For example, the approach in Theorem 1 of factoringa multigraph into hamilton cycles and simple graphs was developed in [9] and used to obtainresults on the chromatic index of multigraphs. In this paper we use Theorem 1 and relevantsimple graph results to obtain new results on the chromatic index, the number of edge-disjointhamilton cycles and the number of p-factors in a regular multigraph of high degree and even order.For the rest of this paper, Theorem Mi will denote the multigraph extension of the simple graphresult denoted by Theorem Si.

2. CHROMATIC INDEX OF MULTIGRAPHS

The chromatic index of a multigraph G, denoted by χ′(G) is the minimum number of colorsneeded to color the edges of G such that no two adjacent edges are assigned the same color. Thefollowing results about the chromatic index are known.

Theorem 2 (Vizing [10]). For any multigraphG,∆(G) ≤ χ′(G) ≤ ∆(G)+µ(G). In particular,if G is simple (i.e., µ(G) = 1), then χ′(G) = ∆(G) or ∆(G) + 1.

Theorem S1 (Chetwynd and Hilton [2]). Let G be a ∆-regular simple graph of even order n.If ∆ ≥ 1

2 (√

7 − 1)n, then χ′(G) = ∆.

We now obtain an extension of Theorem S1 to multigraphs using Theorem 1 given in theintroduction. We need the following classic theorem.

Theorem 3 (Dirac [3]). Let G be a simple graph with order n ≥ 3. If δ(G) ≥ n/2, then G ishamiltonian.

Theorem M1. Let G be a ∆-regular multigraph of even order n with multiplicity µ(G) ≤ r,where r is even. If ∆ ≥ r((

√7 − 1)n/2 + 2), then χ′(G) = ∆(G).

Proof. Let ∆ = rs + q, where 0 ≤ q < r. Since ∆ ≥ r( 12 (√

7 − 1)n + 2), we haves ≥ ( 1

2 (√

7 − 1)n + 2) − q/r > 12 (√

7 − 1)n + 1. Let GS be the simple graph underlying G(i.e., GS is the simple graph obtained by replacing all multiple edges by single edges.) Then,by Theorem 3, GS contains a hamilton cycle and hence (since GS has even order) two edgedisjoint 1-factors. By repeated application of Theorem 3 to GS , we can find q edge-disjoint1-factors F1, F2, . . . , Fq of G. Now, G − F1 − F2 − · · · − Fq is an rs-regular multigraph withs > n/2. Hence, by Theorem 1 in the introduction, there exist r/2 edge-disjoint hamiltoncycles, C1, C2, . . . , Cr/2 (and hence r edge-disjoint 1-factors, M1,M2, . . . ,Mr) in G − F1 −F2 −· · ·−Fq such that E(G−F1 −F2 −· · ·−Fq −M1 −M2 −· · ·−Mr) = E(H1)∪E(H2)∪ · · · ∪ E(Hr), where H1, H2, . . . , Hr are edge-disjoint (s − 1)-regular simple graphs. Sinces − 1 ≥ 1

2 (√

7 − 1)n, Theorem S1 gives that χ′(Hi) = s − 1 for each i = 1, 2, . . . , r. Hence,χ′(G) ≤ q + r + r(s− 1) = ∆ as desired.

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3. EDGE-DISJOINT HAMILTON CYCLES IN MULTIGRAPHS

Jackson [6] proved the following theorem about the number of edge-disjoint hamilton cycles insimple regular graphs.

Theorem S2 (Jackson [6]). Let G be a ∆-regular simple graph of order n ≥ 14. If ∆ ≥(n− 1)/2, then G contains b(3∆ − n + 1)/6c edge-disjoint hamilton cycles.

We obtain the following multigraph extension of Theorem S2.

Theorem M2. Let G be a ∆-regular multigraph of even order n ≥ 14 with multiplicity µ(G) ≤r, where r is even. Suppose ∆ ≥ r(1 + n/2). Write ∆ = rs + q, where 0 ≤ q < r. Then Gcontains bq/2c + r/2 + rb(3(s− 1) − n + 1)/6c edge-disjoint hamilton cycles.

Proof. By repeated application of Theorem 3 to the simple graph underlying G, we can findbq/2c edge-disjoint hamilton cycles H1, H2, . . . , Hbq/2c. If q is even, then G−H1−H2−· · ·−Hbq/2c is an rs-regular multigraph. If q is odd, in addition to bq/2c edge-disjoint hamilton cyclesH1, H2, . . . , Hbq/2c, we can find a 1-factor F in G such that G−H1 −H2 − · · · −Hbq/2c − Fis an rs-regular multigraph. In either case, we are left with an rs-regular multigraph G′, wheres > n/2, i.e., s ≥ 1 + n/2 (since n is even). Hence, by Theorem 1 in the introduction, thereexist r/2 edge-disjoint hamilton cycles C1, C2, . . . , Cr/2 in G′ such that E(G′−C1−C2−· · ·−Cr/2) = E(S1) ∪E(S2) ∪ · · · ∪E(Sr), where S1, S2, . . . , Sr are edge-disjoint (s− 1)-regularsimple graphs. Now, since s − 1 ≥ n/2, by Theorem S2, we can find b(3(s− 1) − n + 1)/6cedge-disjoint hamilton cycles in each Si, 1 ≤ i ≤ r. This gives the required number of edge-disjoint hamilton cycles in G.

4. EDGE-DISJOINT p-FACTORS IN MULTIGRAPHS

Let G be a ∆-regular multigraph of even order n. If there exist ∆ edge-disjoint 1-factors in G,thenG is said to be 1-factorizable. Hilton [5] offered the following weakening of the property of amultigraph being 1-factorizable. For a ∆-regular multigraph of even order n, let (p1, p2, . . . , pk)be a partition of ∆, i.e., pi > 0 for 1 ≤ i ≤ k and p1 + p2 + · · · + pk = ∆. If thereexist k edge-disjoint regular spanning subgraphs of G of degrees p1, p2, . . . , pk, respectively,then G is said to be (p1, p2, . . . , pk)-factorizable. Note that if G is 1-factorizable, then G is(p1, p2, . . . , pk)-factorizable for all partitions (p1, p2, . . . , pk) of ∆. A p-factor of a multigraphG is a p-regular spanning subgraph of G and G is said to be p-factorizable if G contains ∆/pedge-disjoint p-factors. Hilton [5] studied the properties of a simple regular graph of even orderbeing (p1, p2, . . . , pk)-factorizable and beingp-factorizable. In particular, he proved the followingtheorems.

Theorem S3. Let G be a ∆-regular simple graph with even order n ≥ 14. Let (p1, p2, . . . , pk,pk+1, . . . , pl) be a partition of ∆ where p1, p2, . . . , pk are odd positive integers and pk+1, pk+2,. . . , pl are even positive integers. If ∆ ≥ n/2 and k ≤ 2b(3∆ − n + 1)/6c, then G is(p1, p2, . . . , pk, pk+1, . . . pl)-factorizable.

Theorem S4. Let G be a ∆-regular simple graph of even order n and p ≥ 3 be an odd positiveinteger. If ∆ ≥ n/2, and p divides ∆, then G is p-factorizable.

We now obtain extensions of Theorems S3 and S4 to multigraphs. We will need the followingwell-known theorem of Petersen's, and a background result on edge-coloring.

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Theorem 4 (Petersen [7]). Every regular multigraph of even degree is 2-factorizable.

Theorem 5. Let G be a ∆-regular multigraph with even order n ≤ 12 and multiplicity µ(G) ≤r, where r is even and ∆ ≥ rn/2. If n ≤ 8 then χ′(G) = ∆ and if n ≤ 12 then χ′(G) ≤ ∆ + 1.

Proof. A well known lower bound forχ′(G) is given byk(G) = max{2|E(S)|/(|V (S)| − 1)}where the maximum is taken over all induced subgraphsS ofG such that |V (S)| is odd and greaterthan one. Let φ(G) = max{∆(G), dk(G)e}. In [8], it was shown that for any multigraph G oforder n, if n ≤ 8 then χ′(G) = φ(G), and if n ≤ 12 then χ′(G) ≤ φ + 1. However, it is easy tosee that φ(G) = ∆ for the narrower class of regular multigraphs for which ∆ ≥ rn/2 consideredhere (see [9]).

Theorem M3. Let G be a ∆-regular multigraph with even order n and multiplicity µ(G) ≤ r,where r is even. Suppose ∆ ≥ r(1 + n/2). Write ∆ = rs + q, where 0 ≤ q < r. Let(p1, p2, . . . , pk, pk+1, . . . , pl) be a partition of ∆ where p1, p2, . . . , pk are odd positive integersand pk+1, pk+2, . . . , pl are even positive integers. If k ≤ r+ q+2rb(3(s− 1)−n+1)/6c, thenG is (p1, p2, . . . , pk, pk+1, . . . , pl)-factorizable.

Proof. First, suppose that n ≥ 14. Then by Theorem M2, G contains q/2 + r/2 +rb(3(s− 1) − n + 1)/6c edge-disjoint hamilton cycles if q is even and, if q is odd, then G con-tains bq/2c+r/2+rb(3(s− 1) − n + 1)/6c hamilton cycles and one 1-factor, all edge-disjoint.In either case, since n is even, this provides q + r + 2rb(3(s − 1) − n + 1)/6c edge-disjoint1-factors in G. Remove k of these edge-disjoint 1-factors from G to obtain a multigraph G′ thatis (∆− k)-regular. Since ∆ and k have the same parity, ∆− k is even and hence by Theorem 4,G′ is 2-factorizable. Now, the required factorization of G is easily obtained as follows: If pi iseven, then a pi-factor is obtained by combining 1

2pi 2-factors and if pi is odd, then a pi-factor isobtained by combining 1

2 (pi − 1) 2-factors and one 1-factor.Now suppose that n ≤ 12. If n ≤ 8, then by Theorem 5, χ′(G) = ∆. So, G is 1-

factorizable and hence (p1, p2, . . . , pk, pk+1, . . . , pl)-factorizable. If n = 10, then by Theorem 5,χ′(G) ≤ ∆ + 1. Consider a (∆ + 1)-coloring of the edges of G. At least ∆ − 4 colors con-stitute 1-factors in G, else the number of edges in G covered by a (∆ + 1)-coloring is lessthan 5(∆ − 4) + 4(5) = 5∆, which is the number of edges of G. It is easily checked thatk ≤ q + r + 2rb(3(s− 1) − n + 1)/6c with n = 10 and r ≥ 2 implies that k ≤ ∆ − 4. Hence,we remove k 1-factors from G to obtain a multigraph G′ that is (∆ − k)-regular and proceed asin the case when n ≥ 14. The case when n = 12 is similarly handled by using Theorem 5 to firstfind (∆ − 5) edge-disjoint 1-factors in G.

Theorem M4. Let G be a ∆-regular multigraph of even order n and multiplicity µ(G) ≤ r,where r is even. Let p ≥ 3 be an odd integer that divides ∆. If ∆ ≥ r(1 + n/2), then G isp-factorizable.

Proof. Let k = ∆/p and as in the statement of Theorem M3, write ∆ = rs + q where0 ≤ q < r. By Theorem M3, it suffices to show that k ≤ q + r + 2rb(3(s− 1) − n + 1)/6c.Note that k and ∆ have the same parity, and so do ∆ and q since r is even; hence k−r−q is even.Now, since p ≥ 3 and ∆ ≥ r(1+n/2) we have, k− r− q = (∆/p)− r− q ≤ (∆/3)− r− q ≤∆− 2

3r(1+n/2)−r−q = rs−rn/3− 53r < rs−rn/3 = 2[r/6(3(s−1)−n+1)]. Hence, since

k−r−q is even, we have k−r−q ≤ 2b(3(s− 1) − n + 1)/6c ≤ 2rb(3(s− 1) − n + 1)/6c.

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References

[1] J. A. Bondy and U. S. R. Murty, Graph theory with applications, Macmillan, London (1976).

[2] A. G. Chetwynd and A. J. W. Hilton, 1-factorizing regular graphs of high degree—an improved bound,Discrete Math. 75 (1989), 103–112.

[3] G. A. Dirac, Some theorems on abstract graphs, Proc. London Math. Soc. 2 (1952), 69–81.

[4] S. I. El-Zanati, M. J. Plantholt, and S. K. Tipnis, Factorization of regular multigraphs into regularsimple graphs, J. Graph Theory 19 (1995), 93–105.

[5] A. J. W. Hilton, Factorizations of regular graphs of high degree, J. Graph Theory 9 (1985), 193–196.

[6] B. Jackson, Edge-disjoint hamilton cycles in regular graphs of large degree, J. London Math. Soc.(2)19 (1979), 13–16.

[7] J. Petersen, Die Theorie der regularen graphen, Acta Math. 15 (1891), 193–220.

[8] M. J. Plantholt, An order-based bound on the chromatic index of a multigraph, J. Comb., Info. andSystem Sciences 16 (1992), 271–280.

[9] M. J. Plantholt and S. K. Tipnis, Regular multigraphs of high degree are 1-factorizable, J. LondonMath. Soc. (2) 44 (1991), 393–400.

[10] V. G. Vizing, On an estimate of the chromatic class of a p-graph, Diskret. Analiz. [Russian] 3 (1964),25–30.

Received October 28, 1996