perfect path double covers in every simple graph

Perfect Path Double Covers in Every Simple

Hao Li* Graph

LABORATOIRE DE RECHERCHE EN INFORMATIQUE

UNIVERSIT~ DE PARIS SUD ORSAY FRANCE

ABSTRACT

We prove in this paper that every simple graph G admits a perfect path double cover (PPDC), i.e., a set of paths of G such that each edge of G belongs to exactly two of the paths and each vertex of G is an end of exactly two of the paths, where a path of length zero is considered to have (identical) ends. This was conjectured by A. Bondy in 1988.

1. INTRODUCTION AND DEFINITION

In this paper, we consider finite simple graphs and also finite digraphs when specially mentioned. We use the notation and terminology of [2]. For a given graph G, V(G) denotes the vertex set of G and E(G) denotes the edge set of G; Similarly, V ( D ) the vertex set and A(D) the arc set of a digraph D. If P = U ~ U Z . . .up is a path in G, let P[ui,uj] = U , U ~ + ~ . . . u, for any 1 I i I j I p. For a vertex u in a graph G, we denote by Nc(u) the neighbor set of u in G. For two vertices u and u in a digraph D, we denote by (u,u) the arc with tail u and head u, by d i ( u ) the indegree of u, i.e., the number of arcs with head u and by dA(u) the outdegree of u, i.e., the number of arcs with tail u. For a graph G, let n = IV(G)l, rn = (E(G)( and A be the maximum degree of G.

In [l], A. Bondy introduced the following definition:

"Also at Institute of System Science. Academia Sinica. Beijing. China.

Journal of Graph Theory, Vol. 14, No. 6, 645-650 (1990) 0 1990 John Wiley 8i Sons, Inc. CCC 0364-9024/90/060645-06$04.00

646 JOURNAL OF GRAPH THEORY

Definition 1. (1) A path double cover (PDC) of a graph G is a collection II of paths of G such that each edge of G belongs to exactly two paths of II. (2) A small path double cover (SPDC) of a graph G on n vertices is a PDC II of G such that IIIl I n. (3) A perfect path double cover (PPDC) of a graph G is a PDC II of G such that each vertex of G is an end of exactly two paths of n. Note that a path of length zero is considered to have (identical) ends. A. Bondy conjectured in [l] that every simple graph admits an SPDC (The Small Path Double Cover Conjecture) and every simple graph admits a PPDC (The Perfect Path Double Cover Conjecture). It is clear that the SPDC conjecture is weaker than the PPDC conjecture, i.e., the validity of the PPDC conjecture implies the validity of the SPDC conjecture. Until now only a few families of graphs have been proved to admit a PPDC, for instance, K , ([l]), K,,,, ([4]), and the graphs in which every vertex is of odd degree ([3]). In this paper, we shall show that the PPDC conjecture is true, and hence the SPDC conjecture is also true. Therefore, we have

Theorem 1. Every simple graph admits a PPDC.

Theorem 2. Every simple graph admits an SPDC.

In fact, our proof will give a polynomial algorithm in O(mA) steps to find a PPDC for a simple graph.

2. THE PROOF OF THEOREM 1

First Part

We prove Theorem 1 by induction on the number of edges of the graph. First, if the number of edges of the graph is zero, we have a path of

length zero for each vertex of the graph. Clearly, a set of these paths is a PPDC of the graph. Assume that every simple graph with less than m edges admits a PPDC. Then we will prove that every simple graph with m edges admits a PPDC.

Suppose that G is any simple graph with m > 0 edges and e = ab is an edge of G. Then by the induction assumption, there is some PPDC n of G - e. In what follows, we will construct a PPDC of G from n and hence complete the induction.

In a first step we give a transformation on the paths in n to get a set II* of paths of G such that the edge e = ab belongs to exactly one path of II*, each of the edges in E(G) - {e} belongs to exactly two paths of n*, b is an end of exactly one path of II*, a is an end of exactly three paths of II*, and each of the vertices in V ( G ) - {u,b} is an end of exactly two paths of II*.

PERFECT PATH DOUBLE COVERS 647

Then in a second step we apply the same transformation by exchanging the roles of a and b, and so we obtain a PPDC .** of G.

For every vertex u in V(G), let P'(v) and P"(v) be the two paths in Il that end at u. Let NG(u) = (6, U I , u2,. . . , uq}.

Second Part

ideas of Proof

We first give an intuitive version of the proof to show the transformation used. The reader can skip it and go directly to the formal proof followed.

If one of P'(b) and P"(b), say P'(b), does not contain a, then replace P'(b) by P*(b) = P'(b) U {ba} (see Figure 1). Then 11* = [II - {P'(b)}] U {P*(b)} satisfies the desired property and hence the first step is complete.

Suppose now that both P' (b ) and P"(b) contain a. Without loss of generality, let u 1 be such that u l a E P'(b)[b,a], i.e., u I is just before a on the path P'(b) from b to the other end. Then let

P*(b) = [P'(b) - { U l U } ] u {ab}.

This is a path in G that has one end u 1 and contain ab. We now consider P ' ( u l ) and P"(ul). If one of them, say P ' ( u l ) , does not contain a, we put

P * ( U l ) = P'(u1) u (u14 ,

(see Figure 2) and the set 11* = [II - {P'(b), P'(ul)}] U {P*(b), P*(uI ) } , then satisfying the requirement of the first step. If both of P'(u l ) and P"(u 1)

contain a, then at least one of them, say P ' ( u l ) , does not contain the edge u l a since this edge is on P'(b) and belongs to exactly two of the paths in II. Without loss of generality, let u2 be the vertex such that u2a E P'(u l )[u l ,a] , i.e., u 2 is just before a on the path P ' ( u l ) from u I to the other end. Then we put

P * ( U l ) = [ P ' ( U l ) - {uza}l u {ula}.

Then we investigate the two path P ' ( u z ) and P ( u 2 ) , . . . and so on.

minates. For a complete proof, we need the following lemma. The difficulty is to show that the above algorithm of constructing Il* ter-

a

Figure 1


Figure 2

Formal Proof

Lemma 1. such that

Let D be a digraph with vertex set V ( D ) = {x , U I , UZ, . . . , uq, y}

(1) for each u, ,1 I i I q&(u,) = 2 and di (u , ) I 2; (2) d r ; ( x ) - d D ( x ) L 1.

Then there exists a directed path from x toy.

Proof of Lemma I: By contradiction. Suppose that D has no such directed path. Then it is clear that D has no directed walk from x toy.

Let Q be a longest directed walk starting from x. Suppose that u, is the other end of Q, for some j, 1 I j I q. Then we have only two possibilities: (1) exactly one arc incident to u, (with head u,) is in the walk Q, or (2) exactly three arcs incident to u, (two with head u, and one with tail u,) are in the walk Q. But in both the cases, since d;(u,) = 2, there is another arc with tail u, that is not in Q. Thus we can add this arc into Q and obtain a directed walk longer than Q, a contradiction. If Q terminates at x then, since d ; ( x ) - d i ( x ) 2 1, there is some arc with tailx that is not in Q. Sim- ilarly, we can add this arc into Q and get a walk longer than Q, a contradiction, which completes the proof of the lemma. I

Let NG(a) = {b ,u l ,u z , . . . ,uq} . We define a digraph D = (V(D) , A(D)) with V ( D ) = {b ,u l , . . . , uq,m}, (where represents a special vertex) in the following way: for convenience, we put uo = b. Then for each vertex u, , 0 I i I q, consider the two paths P ' ( u l ) and P"(u,) in n starting at u,. If P ' ( u l ) (respectively P"(ul)) contains the vertex a, let u, be the vertex such that u,a E P'(u,)[u,,a], i.e., u, is just before a on the path P' (u l ) from u, to the other end. Then we add to in D an arc (u,, u,). If p ' (u l ) (respectively P"(uJ) does not contain a, then we add an arc (u,, m).

Let us consider the digraph D. Since each vertex is an end of exactly two paths in lI and since each edge of G - e is covered by exactly two paths of II, we have dh(u, ) = 2 and dD(u,) 5 2 for every i, 1 I i I q, and

PERFECT PATH DOUBLE COVERS 649

d g ( b ) = 2 and d;(b) = 0. By Lemma 1, D has a directed path Q from 6 to 0. Assume that this directed path is Q = b u I u z . . .Vim, where the arcs ( b , ~ , ) , ( U , , U , + ~ ) , 1 I i I t - 1, and (u,,m) correspond to the paths P'(b), P'(u,) , 1 5 i I t - 1, and P'(u,) , respectively. Then we put

Po* = [P'(b) - { U l U } u {ab},

P? = [PI( u , ) - {ui+la}] U {.,a}, for every 1 I i 5 t - 1 ,

and

It then follows that each P?,O I i I t, is a path in G that ends at u,+~, for any i,O 5 i 5 t - 1, and that Pi* ends at a.

Put

n* = [n - {P'(ui):O I i 5 t}] u {Pic:O 5 i 5 t} .

Now n* is a collection of paths of G such that each edge of G - e belongs to exactly two paths of n and each vertex of G - {a,b} is an end of exactly two paths of II. Moreover, e = ab belongs to exactly one path P,*, a is an end of exactly three paths of n* and b is an end of exactly one path of n*. Therefore the first step is finished.

Based on II*, by considering the symmetry of a and b, we define a similar digraph D' with vertex set NG(b) U {m}. It may be seen easily that the only differences are d&(a) = 3 and d i , ( a ) I 1. Therefore we may do similar transitions of the paths by using the lemma and get another collection n** of paths in G such that each edge of G - e belongs to exactly two paths of II**; each vertex of G - {a&} is an end of exactly two paths of IT**; e = ab belongs to exactly two paths of II**; and each of a and b is an end of exactly two paths of n**. Hence II** is a PPDC of G and we finish the second step.

The induction is complete and so the theorem is proved. I

Remark. In the above proof, we can find the directed path in D from b to m in O(A) steps. Hence we can find a PPDC of any simple graph G in O(mA) steps.

ACKNOWLEDGMENT

The author thanks J.-C. Bermond, J.-L. Fouquet, F. Jaeger, and A. Ras- paud for their helpful comments.


References

[l] J. A. Bondy, Perfect path double covers of graphs. Personal communi-

[2] J. A. Bondy and U. S. R. Murty, Graph Theory with Applications.

[3] L. Lovasz, On covering of graphs. Theory of Graphs. Academic Press,

[4] K. Seyffarth, personal communication.

cation.

Macmillan Press, London (1976).

New York (1968) pp. 231-236.

perfect path double covers in every simple graph

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