As t 5 m we may use the Konig-Hall Theorem that j=l the collection Cl, Cs, . . . , C, has a distinct set of representatives, say, al, az, , . . , at. Color the edge (ui, U) ...
Appl. Math. Lett. Vol. 7, No. 1, pp. 63-67, 1994 Printed in Great Britain. All rights reserved
Copyright@
Acyclic Edge-Colorings
0893-9659194 $6.00 + 0.00 1994 Pergamon Press Ltd
of Sparse Graphs
Y. CARO Department
of Mathematics,
School
of Education
University
of Haifa-Oranim,
Tivon,
Israel
Y. School
36910
RODITTY
of Mathematical Tel-Aviv,
Sciences, Israel
Tel-Aviv
University
69978
(Received June 1993; accepted July 1993)
Abstract-A
k-forest is a forest in which the maximum degree is k. The k-arboricity denoted Ak(G) is the minimum number of k-forests whose union is the graph G. We show that if G is an m-degenerate graph of maximum degree A, then Ak(G) 5 [(A + (k - 1) m - 1)/k], k 2 2, and derive several consequences of this inequality.
1.
INTRODUCTION
We follow in general the graph terminology and notations of [l]. A k-forest is a forest in which the maximum degree is k. The k-arboricity of a graph G, denoted Ak(G), is the minimum number of k-forests whose union is G. Obviously Al(G) is the so-called chromatic index x’(G) and by Vizing’s Theorem [2], A(G) < Al(G) 5 A(G) + 1. The invariant AZ(G), has been called the linear arboricity of G as discovered in [3]. The main conjecture, due to Akiyama, Exoo and Harary [4], states that if G is a d-regular graph then AZ(G) 5 [(d + 1)/2]. This conjecture is still open, but a breakthrough due to Alon [5] and Guldan [6] shows that the linear arboricity conjecture is valid for graphs with large girth, namely if the girth g(G) 2 14d, and G is a d-regular graph, then AZ(G) 5 [(d + 1)/2]. It was shown in [5] that
As(G) 5 4 + cl A2’3 (log A)“3. Guldan (71 considered the problem of linear arboricity for non-regular graphs with emphasis on almost regular graphs. Recently Kainen [S] considered the linear arboricity of m-degenerate graphs. We shall comment on his results later. Recall that the arboricity of G, A(G), is just the minimum number of forests whose union is G, (which is in our notation A,(G)). The theorem of Nash-Williams [9] states that, A,(G) =
EC%NW4H)
-
1)l.
Another acyclic invariant is the strong chromatic index, d(G), which is the minimum number of colors needed in a proper edge coloring such that every color class is induced, namely, the edge set between any two edges of the color class is empty. A related concept is that of the acyclic edge coloring, at(G), which is the minimum number of colors needed in a proper edge coloring such that each cycle intersects at least three color classes. For results on the strong chromatic index, we refer to [lo]. For results on the acyclic colorings index, we refer to [ll], where it is shown that for every graph UC(G) 5 128A. The authors would like to thank Prank Harary for his valuable remarks.
63
Y. CARO AND Y. RODITTY
64
Our motivation comes from some simple observations:
(1) Most
of the results obtained by using the Lovasz Local Lemma [12] can be improved substantially for m-degenerate graphs.
(2) For
k 2 2, we cannot expect a relation better than Ak(G) 5 [(A + 1)/2], since Ak(G) 1 A,(G) = [(A + 1)/2] ho Ids whenever G is a A-regular graph. On the other hand, if we consider nonregular graphs, then in certain cases we shall prove a bound of the form Ak(G) = A/k + c, where c is a positive constant.
(3) Some
of our results can be implemented by a polynomial time algorithm which is not the case when using the Lovasz Local-Lemma.
An m-degenerate graph is a graph G such that for every subgraph H c G there exists a vertex u E H for which deg, v I m. For some properties of m-degenerate graphs we refer to [13].
2. k-ARBORICITY
OF m-DEGENERATE
GRAPHS
We start with the main theorem of this section. Our method combines counting and an elaboration of an idea used by Kainen [8] who proved the case k = 2. However, his method seems to have difficulties for k 2 3. THEOREM 2.1. If k 2 2 is an integer and G is an m-degenerate graph, then, A+(k-l)m-1
&z(G) 5
k
PROOF. We use induction on the number of vertices of G. The inequality is true for G = Ks. Suppose we have proved it for IV(G) ( = 71. Let G be a graph on n + 1 vertices which is m-degenerate. From the definition there is a vertex v E V(G) such that degG o 5 m and Gr = G-v is m-degenerate on n vertices. Hence, by the induction hypothesis, Ak(G1) < ](A+(k-1)m -1)/k] := y. Let N(v) = {ui : i = 1,. . . , t} be the set of neighbours of v in G. Clearly, t 5 m. Observe that, dega, ui 5 A(G) - 1. Consider a partition (edge-coloring) of Gr into k-forests that realizes Ak(G1). For 1 5 i 5 t and 0 5 j < k define, Ai,j = {all the colors among 1,2, . . . , y that appear in Ui exactly j times}. Put IA,,jI = xi,j. By counting colors and edges we have: k
For 1 I i I t, Cjxi,j
5 A - 1 (counting edges).
(1)
j=O
ForlSilt,
1
A+(k-l)m-1
kZi,j=
k
j=O
(counting colors).
(2)
Multiplying (2) by k and subtracting (1) from (2), we get
&(I: - A xi,j 2 (k - 1) m,
for 1 < i 5 t.
(3)
j=O
Hence, dividing by k - 1 and as k 2 2, we have, k-l
2xi,o + C j=l
“i,j
2 m,
for 1 < i 5 t.
(4
AcyclicEdge-Colorings
65
and set &,s = Ai,o U A:,,.
Now to each member a E Ai,o associate a’ E Ai,,
From (4) we get
k-l
U A,,j U &,o. As t 5 m we may use the Konig-Hall Theorem that ( j=l ) the collection Cl, Cs, . . . , C, has a distinct set of representatives, say, al, az, , . . , at. Color the edge (ui, U) with the color ai, 1 < i 5 t. No color appears in 21more than twice, since all the ai lC’i[ 2 m, where, Ci =
are distinct and as we duplicate only colors of some Ai,o. This event happens only if color a is missing in, say, ‘ZL~ and occur on at most k - 1 edges incident with uj (so that ai = a, a’ = oj) 1 and we obtain the required partition of G using at most [(A + (k - l)m - 1)/k], k-forests. REMARKS.
(1) The
advantage of our proof over that of Kainen is that we avoid case by case consideration, which becomes complicated if k is large. This is mainly due to the counting equations (l)-(4). proof supplies an O(e(G)) algorithm to decompose G into at most ](A+(k-l)m-1)/k]
(2) The
k-forests. Observe that since for a forest F, Ak(G) = [(A(G))/k], we always have the bound (3) w ic is weaker than our result, because of the multiplicative &(G) I Am(G(NG))/kl, h h
(4
factor. Planar graphs are known to be 5-degenerate so that we have for a planar graph G, &(G) I [(A+5(k-1) -1)/k] = [(A-6)/k] +5 for k 2 2 and in particular for the linear arboricity
we get AZ(G) I [A/2] + 2. (5) For 2-degenerate graphs, we get &(G) I [(A + 2(k - 1) - 1)/k] = [(A - 3)/k] + 2, for k > 2 and for k = 2, AZ(G) 5 ](A + 1)/2], which is the conjectured linear arboricity bound. This covers in particular the class of outerplanar graphs, as was also noted by Kainen [8]. (6) Since, Al(G) 5 2A2(G), Th eorem 2.1 supplies a proof that for, e.g., planar graphs x’(G) 2 A + 5 without using the recoloring technique which dominates Vizing’s Theorem. We now apply Theorem 2.1 to graphs with large girth. First we need the following theorem of Bondy and Simonovits [14]. Let k, n and m be positive integers, k 2 2, such that m > 90kn’+““. Then, every graph on n vertices and m edges contains a cycle of length 2t for every k 5 t < knlik. THEOREM A.
THEOREM 2.2. Let G be a graph on n vertices
t nl/“f(n),
where, f(n)
with girth, g(G) 2 2t + 1, t 2 2, and A(G) -+ 00. Then, for a fixed k 2 2, &(G) I $ (1 + o(1)).
1
PROOF. Since g(G) 2 2t + 1 it follows from Theorem A that e(G) 5 90tn’+‘it. Hence, for the average degree d(G) I 80t nllt and since g(H) > 2t + 1 holds for every subgraph H of G, we conclude that G is 180t nllt-degenerate. Then by Theorem 2.1,
&(G)
5 and A L 2n1i2 f(n)
where f(n)
Then for every k 2 2, Ak(G) < $ (1 + o(1)). COROLLARY 2. Let G be a graph on n vertices
f(n)
such that g(G) 2 log n and A 2 g(G) f(n) + co. Then for every k 2 2, Ak(G) 5 p (1 + o(l)).
-+ 00. I where
I As it is well known [13], there exists A-regular graphs with arbitrarily large girth. For such graphs &(G) 1 A,(G) = [(A f 1)/2], hence in some sense Corollary 2 is best possible, because (as pointed by Alon) without any relation between the girth and the order of the graph the arboricity bound [(A + 1)/2], can be realized. I REMARK.
66
Y. CAR~ AND
3. UC(G), d(G) AND
Y.
RODITTY
THE GENUS
OF A GRAPH
Recall first that in (111 it was proved that at(G) I 128A. It is trivial to see that in an mdegenerate graph G we have in fact, W(G) 5 st(G) I m A(G). Our goal is to improve upon this trivial bound, by extending an idea of [12]. Let G be a graph and e = (u, v) be an edge of G. By contracting G at e we mean the graph H obtained from G by replacing the vertices u and v by a single vertex w such that N(w) = N(u) U N(v)
- {u,v},
(see e.g., [13]).
FACT 1. If the genus of G is k >_ 0 and H is a graph obtained by contractions from G, then the genus of H is at most k (see 1151). FACT 2. Let the genus of G be r(G)
= y 1 0 then by the celebrated Heawood-fingel-Young
Theorem and the Four Color Theorem x(G) I [(7 + (1 + 48~)‘/~)/2J. THEOREM 3.1. Let G beagraph
ofgenusy.
(see [13,15]).
Then, at(G) 5 A(G) I ~‘(G)[(7+(1+48y)l/~)/2].
PROOF. Consider an edge coloring using x’ colors. For each color class Ei, 1 < i 5 x’ let Gi be the subgraph G spanned by the edges of Ei. Thus, Ei is a perfect matching in Gi and obviously y(Gi) 5 y(G) = y. Consider Hi the graph obtained from Gi upon contracting all the edges of Ei. By Fact 1, y(Hi) 5 y(Gi) 5 y and two vertices in Hi are independent iff the corresponding edges in Gi form an induced 2K2. By Fact 2, x(Hi) 5 [(7 + (1 + 48~)~/~)/2], which in turn induces a coloring of Ei such that two edges of the same color form an induced 2Kz in G. Hence, at(G) I St(G) 5 x’(G) [(7 + (1 + 48-~)~/~)/2] I (A + 1) l(7 + (1 + 48~)~/~)/2J. REMARKS. (1) For a planar graph G we obtain at(G) 5 St(G) 5 4x’, which gives at(G) < 4A providing A 2 8, since in this case we have from the theorem of Vizing x’(G) = A. This is a slight improvement over the result obtained in [lo], where the bound 4A + 4 is established. (2) The upper bound in Theorem 3.1 is better than the probabilistic bound as long as y 2 1250. (3) For outerplanar graphs Theorem 3.1 gives only at(G) I St(G) 5 4A, and with a little more effort using forbidden minors in outerplanar graphs this bound can be reduced to 3A. However, the next theorem shows that quite often this bound is still far from the truth. Define the mtimzlm
edge-connectivity,
X(G) = 11u~~~,X(u,v),
where, X(u, v) is the edge-
connectivity of the pair U, v. This parameter is studild in [13]. THEOREM 3.2. Let G be a 2-degenerate
graph with X(G) = k. Then, at(G) 5 A + k - 1.
PROOF. The claim holds for small graphs. Let G be a 2-degenerate graph with X(G) = k. Let v E V(G) be a vertex of degree 2, (if degv 2 1 we have nothing to prove), then H = G - v is also 2-degenerate with X(H) 5 i(G) = k. By the induction hypothesis at(H) 5 A(H) + k - 1 5 A(G) + k - 1. Let U, w be vertices adjacent to v in G and let C(w), C(U) be the set of colors not used on edges incident with w respectively, u, in H. As deg, w, deg, u 5 A - 1 it follows that IC(w)l,IC(~)I 2 k. W e may assume that k 2 2 since X(G) = 1 implies that G is a forest and there is nothing to prove in that case. Suppose a E C(U), b E C(w), a # b, then we can color (u, v) by a and (v, w) by b and this is a proper coloring unless b $ C(u) a 4 C(w) and there is a path with alternating colors a, b from u to w. As IC(w)I, [C(U)[ 2 k 2 2 and by the above argument we infer that C(U) n C(w) = 0, the coloring above applies unless there are k edge-disjoint paths from u to w with the suitable color pairs which together with the path u-v-w give X(u, w) 2 k + 1, a contradiction. I
Acyclic
Edge-Colorings
67
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