National Sun Yat-sen University Institutional Repository:Item 987654321/39811

行政院國家科學委員會專題研究計畫 期中進度報告

圖的圓色數及應用(2/3)

中 華 民 國 94 年 5 月 17 日

The circular chromatic index

of graphs of high girth

∗

Tom´

aˇs Kaiser

Daniel Kr´

al’

Riste ˇ

Skrekovski

Xuding Zhu

Abstract

We show that for each ε > 0 and each integer ∆≥ 1, there exists a number g such that for any graph G of maximum degree ∆ and girth at least g, the circular chromatic index of G is at most ∆ + ε.

1

Introduction

A proper edge-coloring of a graph G is a coloring of all the edges of G such that every two incident edges receive distinct colors. The smallest number of colors for which there is a proper edge-coloring is called the chromatic index and denoted by χ
(G). Recall that the chromatic index of a simple graph

∗_{The first three authors are supported by the Czech-Slovenian bilateral projects Mˇ}

SMT-07-0405 and SLO-CZ/04-05-002. The last author is supported by the National Science Council (Taiwan) grant NSC93-2115-M-110-004.

†_{Department of Mathematics and Institute for Theoretical Computer Science (ITI),}

University of West Bohemia, Univerzitn´ı 8, 306 14 Plzeˇn, Czech Republic. E-mail: [email protected]. Institute for Theoretical Computer Science (ITI) is supported by Ministry of Education of Czech Republic as project LN00A056.

‡_{Department of Applied Mathematics and Institute for Theoretical Computer Science}

(ITI), Charles University, Malostransk´e n´amˇest´ı 25, 118 00 Praha 1, Czech Republic. E-mail: [email protected].

§_{Department of Mathematics, University of Ljubljana, Jadranska 19, 1111 Ljubljana,}

Slovenia. E-mail: [email protected].

¶_{Department of Applied Mathematics, National Sun yat-sen University,}

Kaohsi-ung, Taiwan 80424, and National Center for Theoretical Sciences, Taiwan. E-mail: [email protected].

with maximum degree ∆ is either ∆ or ∆ + 1 by the classical theorem of Vizing [24]. A graph is said to be of Vizing class I if its chromatic index is ∆; all other graphs are of Vizing class II. In this paper, we study circular edge-colorings of graphs with large girth and we show that their circular chromatic index is close to ∆.

The concept of circular coloring of graphs was introduced by Vince [23]. A proper circular -coloring of a graph G, for a real ≥ 1, is a coloring of the vertices of G by real numbers from the interval [0, ), such that the difference modulo  of the colors γ₁ and γ₂ assigned to two adjacent vertices is at least one, i.e., 1≤ |γ₁− γ₂| ≤  − 1. A circular -coloring can also be viewed as a coloring by points on a circle of circumference  in such a way that a pair of adjacent vertices receive colors which are at distance at least 1 on the circle. The smallest real  for which there is a proper circular -coloring is called the circular chromatic number of G and denoted by χ_c(G) (the minimum is always attained and it is a rational number [5, 23]). It can be shown that χ(G)−1 < χ_c(G)≤ χ(G). For further results on circular coloring, the reader is referred to the survey [27] on the subject.

A circular -edge-coloring of a graph G is just a circular -coloring of the line graph L(G). The circular chromatic index χ
(G) of G is defined to be the circular chromatic number of L(G). It is not difficult to show that ∆≤ χ
_c(G)≤ ∆ + 1 for every graph G with maximum degree ∆.

Jaeger and Swart [12] conjectured that there is an integer g such that each snark has girth at most g (actually, they conjectured that g = 6 suffices). Recall that the girth of a graph is the length of its shortest cycle, and that a snark is a cubic bridgeless graph belonging to the Vizing class II [6, 25] (i.e., of chromatic index 4). We remark that the definition of snarks is often taken to include further assumptions (as, e.g., in [14]) to avoid trivial cases.

The conjecture of Jaeger and Swart has become known as the Girth con-jecture. More than 15 years after it was stated, the conjecture was disproved by Kochol [14] who constructed cyclically 5-edge-connected snarks with arbi-trarily large girths. The authors [13] recently showed that the Girth conjec-ture is true when relaxed to the circular edge-coloring in the following sense: for each ε > 0, there is a number g such that the circular chromatic index of any cubic bridgeless graph of girth at least g is at most 3 + ε.

In the present paper, we extend this result to graphs of higher degree. This provides an affirmative answer to the question presented in [13, Problem 2]: is it true that for any ∆ and ε > 0, there exists an integer g such that every ∆-regular graph G of girth at least g has χ
_c(G)≤ ∆ + ε? Actually, we

prove the result for graphs of maximum degree ∆ (which are not necessarily ∆-regular).

In general, large girth does not imply that the circular chromatic number of a graph is small. For any integers k ≥ 1 and g ≥ 3, there is a graph G with girth at least g, and with χ(G) = χ_c(G) = k as shown by Steffen and Zhu [21]. On the other hand, Galluccio, Goddyn and Hell [10] proved the following for graphs avoiding a fixed graph H as a minor: For every graph H and every ε > 0, there is an integer g such that the circular chromatic number of each graph whose girth is at least g and which does not contain H as a minor is at most 2 + ε.

A well-known question related to the girth and the circular chromatic number is the Pentagon problem (see, e.g., [18]): does every cubic graph G of sufficiently large girth admit a homomorphism into the 5-cycle? (Recall that a homomorphism from G to H is a mapping from V (G) to V (H) carrying edges of G to edges of H.) It is not hard to see that the above homomorphism exists if and only if the circular chromatic number of G is at most 5/2. Let us stress, however, that in the present paper, we are concerned with the circular chromatic index, i.e., the circular chromatic number of the line graph. Thus, our results have no direct implications for the Pentagon problem.

Similarly as in [13], the proof of our main result uses the concept of independent systems of representatives [2–4, 9, 11, 16, 17]. The technique used in [13] can be easily generalized to the case of ∆-regular (∆− 1)-edge-connected graphs. However, when the connectivity assumption is removed, the technique fails to work. To be able to use it, one must find a way to get rid of small edge-cuts.

Small edge-cuts may indeed be dangerous: for instance, Afshani et al. [1], motivated by the question of Zhu [27, Question 8.4], showed that the circular chromatic index of any cubic bridgeless graph is at most 11/3. However, their result does not hold for graphs with bridges (nor, for that matter, for bridgeless graphs of maximum degree 3). In addition, it is known [8, 20] that the so-called fractional chromatic index χ
_f(G) of every ∆-edge-connected ∆-regular graph G of even order is equal to ∆, but the statement is false without the connectivity assumption.

Our results imply an upper bound to χ
_f(G), since χ
_f(G)≤ χ
_c(G) (see [27] for details). Specifically: for each ∆ and each ε > 0, there exists an integer g such that the fractional chromatic index of every graph with maximum degree ∆ and girth at least g is at most ∆ + ε (Corollary 16). However, this result on the fractional chromatic index can also be easily proved directly

without using our results on the circular chromatic index.

The outline of the paper is as follows. In Section 3, we show, in a series of lemmas, that the edge set of any cyclically ∆-edge-connected graph of maximum degree ∆ can be decomposed into ∆/2 factors of maximum degree at most 2, one of which is a matching if ∆ is odd. In Section 4, we recall our major tool, a theorem on independent systems of representatives, and some related results. Section 5 contains a proof of a crucial extension property for circular edge-colorings of trees. This is used in Section 6 to prove a special case of our main result for the class of ∆-decomposable graphs, but in a stronger form that allows a subgraph to be precolored. The general case is proved in Section 7.

2

Notation

We use standard terminology and notation of graph theory. For background information on the subject, consult, e.g., [7, 26]. The symbols V (G) and E(G) denote the vertex set and the edge set of a graph G, respectively. The order of G is the number of its vertices, and the size of G is the number of its edges. The result of the contraction of a subgraph H of G is denoted by G/H; this may introduce multiple edges, but any loops that arise are discarded. A matching in G is a set of pairwise disjoint edges. For an integer k, a k-factor of G is a spanning subgraph of G in which each vertex has degree precisely k.

The distance of two edges e and e
is the length of a shortest path whose first edge is e and whose last edge is e
decreased by one. In particular, edges at distance one are incident edges. If H is a subgraph of a graph G and H contains at least one edge, then the distance of an edge e ∈ E(G) from H is the distance of e from the closest edge of H. We write N (H, k) for the subgraph induced by all the edges whose distance from H is at most k. We also use N (H, k) if H is just a set of edges, and we abbreviate N ({e}, k) as N (e, k). The neighborhood N (H) of a subgraph H ⊆ G is defined to be N (H, 1). Finally, V_d(G) stands for the set of all the vertices of a graph G whose degree in G is d.

3

Decompositions of graphs

In this section, we introduce the notions of decomposable graphs and graph decompositions which play a key role in our arguments.

Let G be a graph of maximum degree ∆. For an integer k, a sub-k-factor of G is any spanning subgraph of maximum degree at most k. Note that a 1-factor of graph is a perfect matching, while a sub-1-1-factor is just a matching. An -decomposition of a graph G is a decomposition of G into /2 edge-disjoint sub-2-factors, one of which is required to be a matching if  is odd. If G has an -decomposition, then it is said to be -decomposable. For our purposes, ∆-decomposable graphs of maximum degree ∆ will be of interest. It is easy to find examples of graphs that are not of this type (consider any cubic graph with no perfect matching). However, graphs of even maximum degree always have a ∆-decomposition:

Lemma 1 Every graph of maximum degree 2k, k ≥ 1, has a

2k-decomposi-tion.

Proof: Let G be a graph of maximum degree 2k. It is easy to see that one

may add vertices and edges to G so as to obtain a 2k-regular multigraph G
. By [19] (see also [7, Corollary 2.1.5]), the edges of G
can be decomposed into k 2-factors. This yields a 2k-decomposition of G.

Let G be a connected graph. An edge-cut in G is a minimal set T of edges of G such that G\ T is disconnected. Note that if T is an edge-cut, then G\ T has exactly two components. The graph G is k-edge-connected if it contains no edge-cut of size at most k− 1. An edge-cut T is essential if each component of G\ T contains at least one edge. The graph G is essentially k-edge-connected if|E(G)| > k and G does not contain any essential edge-cut of size at most k− 1. An edge-cut T is cyclic if each component of G \ T contains a cycle. As before, G is cyclically k-edge-connected if |E(G)| > k and G contains no cyclic edge-cut of size at most k− 1.

In the rest of this section, we prove an analogue of Lemma 1 for graphs of odd maximum degree ∆, provided that they are cyclically ∆-edge-connected. The main step is to prove the existence of a matching which covers all vertices of degree ∆.

Theorem 2 A multigraph G contains a 1-factor if and only if for each S ⊂

V (G), the following holds:

c_odd(G\ S) ≤ |S|,

where c_odd(H) denotes the number of components of H of odd order.

The proof of the following lemma follows ideas used to prove the theorem of Petersen [19] on the existence of 1-factors in bridgeless cubic graphs:

Lemma 3 Let G be a connected multigraph. Suppose that all the vertices of

G have the same odd degree ∆ ≥ 3, except possibly for one vertex u of degree at most ∆. If G does not contain any edge-cut of size less than ∆, with a possible exception of the cut formed by all the edges incident with u, then G contains a matching that covers V_∆(G).

Proof: We distinguish two cases regarding the parity of the order of G.

Assume first that |V (G)| is even. We use Theorem 2 to show that G contains a 1-factor. Consider a subset S ⊂ V (G). If S = ∅, then c_odd(G\ S) = 0 as G is connected. Otherwise, the number of edges between the vertices of S and V (G)\ S is at most ∆|S|, since each vertex of S has degree at most ∆. On the other hand, G contains at most one edge-cut of size less than ∆, and so all but at most one component of G\ S are joined to the vertices of S by at least ∆ edges. In particular,

∆(c_odd(G\ S) − 1) < ∆|S|,

which implies c_odd(G\ S) ≤ |S| as required. By Theorem 2, G has a 1-factor. Assume now that |V (G)| is odd. Let δ be the degree of u. Since ∆ is odd, δ < ∆ by the hand-shaking lemma. Thus, it is enough to show that the graph G
= G\ {u} has a 1-factor. Note that G
is connected: otherwise, a proper subset of the edges incident with u would form an edge-cut in G, which was assumed not to be the case.

We again use Theorem 2 in order to show the existence of a 1-factor in G
. Let S
⊂ V (G
) and set S = S
∪ {u}. If S
=∅, then S = {u} and the graph G\ S = G
\ S
is connected. Since its order is even, the condition of Theorem 2 is satisfied. If S
= ∅, then the number of edges between S and V (G)\S is at most ∆|S
|+δ. Each component C of G\S is joined by at least ∆ edges to the vertices of S. Indeed, if not, then the edges incident with C

form an edge-cut of size smaller than ∆. By the assumption of the lemma, the edges forming this edge-cut must be exactly all the edges incident with the vertex u. Then, the graph G[C ∪ {u}] is a component of the graph G, but since G is connected, we infer that S
= ∅. Hence, each component of G\ S is joined by at least ∆ edges to the vertices of S. Since there are at most ∆|S
| + δ < ∆(|S
| + 1) such edges, the graph G \ S = G
\ S
consists of at most |S
| components. In particular, c_odd(G
\ S
)≤ |S
|, and G
contains a 1-factor by Theorem 2.

As announced before, we now strengthen Lemma 3 to the class of essen-tially ∆-edge-connected graphs:

Lemma 4 Every essentially ∆-edge-connected multigraph G of odd

maxi-mum degree ∆ ≥ 3 has a matching which covers all the vertices of V_∆(G).

Proof: As long as G contains a pair of non-adjacent vertices v and v

whose degrees sum up to at most ∆, identify v and v
(preserving multiple edges if they arise). Just like G, the resulting multigraph G
is essentially ∆-edge-connected.

If G
contains a pair of vertices u and v both of whose degrees are smaller than ∆, add the edge uv to G
. (This preserves the essential edge-connectivity since the sum of the degrees of u and v is larger than ∆.) Repeat this process until there is no such pair of vertices, and call the resulting multigraph G

. If G

is ∆-regular, it is also ∆-edge-connected. Otherwise, G

contains exactly one vertex z of degree smaller than ∆, and the only edge-cut in G

of size less than ∆ consists of the edges incident with z. In each case, Lemma 3 implies that G

has a matching which covers V_∆(G

). The matching consisting of the corresponding edges in G has the required property.

Before we handle cyclically ∆-edge-connected graphs, we need to observe the following property of matchings in trees:

Lemma 5 Let T be a tree of maximum degree ∆ and X ⊆ V (T ). Set T_∆= V_∆(T )\ X. If no vertex of T_∆ has ∆ neighbors in X, and at most one vertex of T_∆has ∆−1 neighbors in X, then there is a matching M in T \X covering all vertices of T_∆.

Proof: We can assume that T_∆ is non-empty, for otherwise the claim holds for trivial reasons. Let r be a vertex of T_∆ with ∆− 1 neighbors in X, if there is one. Otherwise, choose r to be an arbitrary vertex of T_∆. Orient the edges of the tree T away from the vertex r.

Let M₀ be a set of edges obtained by choosing, for each vertex v ∈ T_∆, one outgoing edge vw ending in a vertex w /∈ X. Such a choice can always be made, since if v = r, then at most ∆ − 2 out of the ∆ − 1 outgoing edges end in X (and a similar argument applies to r).

Finally, we change M₀ into a matching M by removing certain edges. For every maximal directed path P in M₀, remove every second edge (starting with the second one from the beginning of P ). Clearly, M is a matching, and since each P ends in a vertex not contained in T_∆, M still covers all the vertices of T_∆.

We are now able to extend Lemma 4 to cyclically ∆-edge-connected graphs:

Lemma 6 For any odd integer ∆ ≥ 3, every cyclically ∆-edge-connected

graph G of maximum degree ∆ has a matching covering V_∆(G).

Proof: We proceed by induction on the size of G. If G is essentially

∆-edge-connected, then the claim follows from Lemma 4. Thus, let C be an essential edge-cut of size at most ∆−1; by the cyclic connectivity assumption, at least one of the components A and B of G− C is a tree. We may assume that it is the component A.

Since the multigraph G/A is cyclically ∆-edge-connected and its maxi-mum degree is at most ∆, G/A has a matching M_B covering V_∆(G/A) by the induction hypothesis (note that the claim is trivially true if the maximum degree of G/A is smaller than ∆). We will use the same symbol M_B for the matching in G comprised of the corresponding edges.

Let T be the tree obtained from G/B by splitting the vertex corresponding to B into |C| vertices of degree one. We define X ⊂ V (T ) to be the set consisting of the counterparts of the vertices covered by M_B in G. Observe that |X| ≤ ∆, and if |X| = ∆, then M_B uses an edge of C, and thus X contains two adjacent vertices. It follows easily that the hypothesis of Lemma 5 is satisfied. Hence, there is a matching M_A in T \ X covering

V_∆(T )\ X. Again, we use M_Afor the corresponding matching in G. Clearly, M_A∪ M_B is a matching which covers V_∆(G).

We use Lemma 6 to find a ∆-decomposition of any cyclically ∆-edge-connected graph:

Proposition 7 Any cyclically ∆-edge-connected graph G of maximum degree

∆ is ∆-decomposable.

Proof: If ∆ is even, then G is ∆-decomposable by Lemma 1. For ∆ = 1, the

assertion is trivial. Thus, we may assume that ∆ ≥ 3 is odd. By Lemma 6, the graph G contains a matching M which covers V_∆(G). In particular, G\ M has even maximum degree ∆ − 1. Again by Lemma 1, G \ M has a (∆− 1)-decomposition. Adding M to this (∆ − 1)-decomposition, we obtain a ∆-decomposition of G.

4

Independent systems of representatives

An important ingredient of our proof is the concept of an independent system of representatives, introduced by Fellows [9]. Throughout this section, let G be a graph with a given partition P = {V₁, . . . , V_k} of its vertex set V into k disjoint subsets. A set X ⊆ V is an independent system of representatives (ISR) with respect to P if X is an independent set intersecting each member of P in a single vertex.

Recently, there has been a surge of interest in ISRs, preceded by a result of Haxell [11] which we describe in more detail. Let us start with some terminology. A set D of vertices of the graph G is totally dominating if each vertex of G has a neighbor in D. The size of a smallest totally dominating set in G is the total domination number ˜γ(G) of G. For I ⊆ {1, . . . , k}, set

V_I =

i∈I

V_i.

Finally, if X ⊆ V (G), then G[X] denotes the induced subgraph on the set X. The above mentioned result of Haxell [11] is the following:

Theorem 8 Let G be a graph andP = {V₁, . . . , V_k} a partition of V (G). If, for each I ⊆ {1, . . . , k}, it holds that

˜

γ(G[V_I])≥ 2|I|, then G has an ISR with respect to P.

In the present paper, we make use of the following result which follows from Theorem 8 as a simple corollary:

Theorem 9 Let G be a graph of maximum degree ∆ and letP be a partition

of V (G). If each member of P contains at least 2∆ vertices, then G has an ISR with respect to P.

Although Theorem 9 is sufficient for our needs, we briefly sketch the subsequent development in the area which started with an extension of Hall’s matching theorem to hypergraphs, proved by topological means by Aharoni and Haxell [4]. The result was further generalized by Meshulam [16], and a more elementary proof of the generalization was given by Aharoni et al. [3]. As observed by Ron Aharoni (see [16]), the method of [4] implies a funda-mental result on ISRs that underlies a number of known sufficient conditions including Theorem 8. We refer the reader to [2] for a discussion of these, and for the topological definitions. Here, we just state the result:

Theorem 10 Let G be a graph and P = {V₁, . . . , V_k} a partition of V (G). If the topological connectivity of the independence complex of G[V_I] is at least |I| − 2 for each I ⊆ {1, . . . , k}, then G has an ISR with respect to P.

In [17], the relation between the properties of the independence complex (in particular, its homology) and the existence of an ISR is elaborated even further.

5

Local modifications of edge-colorings

In this section, we show that if e is an edge of a tree T of maximum degree ∆, then any ∆-edge-coloring of T\ e can be modified to a circular (∆ + ε)-edge-coloring c
of T (where ε > 0 is a given real number) such that c
has any prescribed value on e, and c
agrees with c at all edges that are sufficiently far apart from e.

Lemma 11 Let T be a tree of maximum degree ∆ and e ∈ E(T ). Suppose

that c is a proper ∆-edge-coloring of T \ e. Furthermore, let real numbers ε > 0 and γ ∈ [0, ∆ + ε) be given. There exists a circular (∆ + ε)-edge-coloring c
of T such that c
(e) = γ, and for any edge f at distance at least ∆2_{/ε + 2∆ from e,}

c
(f ) = c(f ).

Proof: First, note that it may be assumed that e is incident with a leaf of

T . Indeed, if this is not the case, then let T₁ and T₂ be the trees obtained by adding the edge e to each component of T \ e, apply the statement of the lemma to T₁ and T₂ separately, and combine the corresponding (∆ + ε)-edge-colorings to obtain c
.

By our assumption, e is only incident with at most ∆− 1 other edges of T . Without loss of generality, we may assume that none of these ∆− 1 edges is colored 0. By setting c(e) = 0, c is extended to a proper ∆-edge-coloring of T .

If f is an edge of T , d(f ) denotes the distance between f and e. Let C be a circle of circumference ∆ + ε. We identify the points of C with numbers from [0, ∆+ε) and assume the numbers increase as one traverses C clockwise. A palette is any ∆-tuple

P = (P (0), . . . , P (∆− 1)) ⊂ C

such that for any distinct x and y, the distance between P (x) and P (y) (on C) is at least 1.

Set D = γ∆/ε . In the sequel, we shall define palettes P₀, . . . , P_D+∆. For each edge f ∈ E(T ), the value of c
on f will be

c
(f ) =      γ if f = e, P_{d(f )}(c(f )) if 1≤ d(f) ≤ D + ∆, c(f ) otherwise. (1)

The palette P₀ consists of ∆ consecutive, equally spaced points of C, the first of which is γ. In symbols,

P₀(x) = xλ + γ,

where λ = 1 + ε/∆ and (as in all that follows) all arithmetic is performed modulo ∆ + ε. For i < D, the palette P_i+1 is obtained from P_i by a counter-clockwise rotation by ε/∆. Using (1), this determines the values of c
on all edges at distance at most D from e.

Let us check at this point that the restriction of c
to the edges at distance at most D from e is a proper circular (∆ + ε)-coloring. Since the distances of colors in a palette are at least 1, the only conflict that can occur is that for some distinct x and y in {0, . . . , ∆ − 1} and some i, the distance between P_i(x) and P_i+1(y) on C is less than 1. If this happens, then clearly y = x + 1 (modulo ∆), and thus

P_i+1(x + 1)− P_i(x) < 1. However, one easily calculates that

P_i+1(x + 1)− P_i(x) =  (x + 1)λ + γ− (i + 1)ε ∆  −xλ + γ− iε ∆  = λ− ε ∆ = 1. Hence, we do have a proper coloring so far.

Observe that P_D(0)∈ [0, ε/∆). More generally, P_D(x) is close to xλ: P_D(x) = xλ + γ− _γ∆ ε · ε ∆ ∈ xλ, xλ + ε ∆  . (2)

Next, we “round the colors down”. The ultimate outcome will be the palette P_D+∆ = (0, 1, . . . , ∆−1). The intermediate palettes P_D+1,. . . ,P_D+∆−1 are defined as follows. For i = 1, . . . , ∆, we let

P_D+i(x) = 

x if 0≤ x < i,

P_D(x) otherwise. (3) Note that P_D+1 equals P_D except for the value at 0. Furthermore, the dis-tance of P_D+1(0) from P_D(∆− 1) is at least 1. Hence, c
is proper on all edges in N (e, D + 1).

In fact, c
is proper on all of N (e, D + ∆). To see this, it suffices to check that the distance between P_D+i(x) and P_D+i+1(x + 1) (where 0≤ i < ∆ and 0≤ x < ∆) is at least 1. By (3), this distance is either the distance between x and x + 1 (which is 1), or the distance between P_D(x) and P_D(x + 1) (which is λ > 1), or the distance between x and P_D(x + 1) (which is at least (x + 1)λ− x > 1).

Since P_D+∆(x) = x for all x, the coloring c
coincides with c on edges at distance at least D + ∆ from e. The estimate on the distance from the statement of the lemma follows from

D + ∆ ≤ γ∆ ε + ∆ < (∆ + ε)∆ ε + ∆ = ∆2 ε + 2∆.

6

Coloring

∆-decomposable graphs

For an integer d, a subgraph H of G is said to be d-closed if it contains every path P in G such that P joins two vertices from V (H) and the length of P is at most d.

Theorem 12 For any integer ∆ ≥ 3 and real ε > 0, there exist positive

integers d and g such that the following holds: let G be an arbitrary ∆-decomposable graph of maximum degree ∆ and girth at least g. If H is a d-closed subgraph of G with 1≤ |E(H)| ≤ d∆, then any proper (∆ + ε)-edge-coloring h of the neighborhood of H extends to a (∆ + ε)-edge-ε)-edge-coloring of the entire graph G.

Proof: Set  = ∆2_{/ε + 2∆, d = 2 + 2 and g = 2(d∆ + 1)∆}2
+1_{. We are}

given a ∆-decomposition

F = (F0, . . . , F∆/2−1)

of G, where F_∆/2−1 is a matching if ∆ is odd. A key idea of the proof is to use the ∆-decomposition to construct a ∆-edge-coloring c of G\ S, where S will be a set of edges that are “sparse” and, in addition, distant from the subgraph H. Applying Lemma 11 to a suitable neighborhood of each edge of S∪ ∂H, where ∂H is the set of edges with precisely one endvertex in H, one can then modify c to a (∆ + ε)-edge-coloring of the whole graph G which extends the coloring h.

Let C₁, . . . , C_k be all the odd cycles contained in F₀, . . . , F_∆/2−1. Con-struct an auxiliary graph ˜G whose vertices are all edges e contained in the cycles C_i such that the distance of e from H is at least 2 + 2. Two such edges e and f are joined by an edge in ˜G if they come from different cycles C_i, and their distance in G is at most 2. The vertex set of ˜G is naturally partitioned into k classes ˜C_i corresponding to the cycles C_i.

In a graph of maximum degree ∆≥ 3, the number of edges at distance less than t from any given edge does not exceed

1 + 2(∆− 1) + 2(∆ − 1)2+· · · + 2(∆ − 1)t−1< 2∆t−1 < ∆t.

Consequently, the maximum degree of ˜G is at most ∆2
+1, and there are at most 2d∆2
+2 _{edges in N (H, 2 + 1). Thus, the size of each ˜C}

i is at least

We aim to use Theorem 9 for ˜G. By the choice of g, the hypotheses of Theorem 9 are satisfied, and hence ˜G has an independent system of repre-sentatives S ={e₁, . . . , e_k} with respect to the parts ˜C_i.

Define an edge-coloring c of G\ S as follows. Since S intersects all odd cycles contained in the sub-2-factors of the decomposition F, each graph F_i\ S (where i = 0, . . . , ∆/2 − 1) is bipartite and can be colored by colors 2i and 2i + 1. If ∆ is odd, then F_∆/2−1 is a matching and all of its edges can be colored with the color ∆− 1. Clearly, c is a proper ∆-edge-coloring of G\ S.

Let ∂H be the set of edges of N (H) not contained in H. For each e∈ ∂H, let T₀(e) be the subgraph of G induced by e and all edges in N (e, ) outside H ∪ ∂H, and set T (e) to be the component of T₀(e) containing e. For an edge e∈ S, let T (e) = N(e, ). Since the girth of G is larger than 2( + 1), each subgraph T (e), e∈ ∂H ∪ S, is a tree. Moreover, for two distinct edges e₁, e₂ ∈ ∂H ∪ S, the trees T (e₁), T (e₂) are edge-disjoint. To prove this, it is enough to show that

the distance of e₁ and e₂ is at least 2 + 1. (4) We distinguish three cases. If e₁, e₂ ∈ S, then (4) follows from the fact that S is an ISR for ˜G and the parts ˜C_i. If e₁ ∈ S and e₂ ∈ ∂H (or vice versa), (4) is ensured by the construction of ˜G: the edges of distance less than 2 + 1 from ∂H are not included in ˜G at all. Finally, if both e₁ and e₂ are in ∂H, then (4) holds since H is d-closed.

We now define a (∆ + ε)-edge-coloring h
of G that extends the given edge-coloring h of the neighborhood of H. For each edge e ∈ ∂H ∪ S, use Lemma 11 to find a (∆ + ε)-coloring h
_e of T (e) that agrees with c on the edges of the tree T (e) incident with leaves, and such that

h
_e(e) = 

h(e) if e∈ ∂H, 0 if e∈ S. Now define, for all edges f of G,

h
(f ) =     

h
_e(f ) if f is contained in some tree T (e), h(f ) if f ∈ E(H),

It is straightforward to check that h
is a proper (∆ + ε)-edge-coloring which extends h.

7

The main result

Before we are able to prove our main result, Theorem 15, we must establish an auxiliary lemma on cyclic edge-cuts in graphs.

Lemma 13 Let k ≥ 2 be a given integer and let G be a connected graph with

more than k edges. If G is not cyclically k-edge-connected, then it contains a cyclic edge-cut of size at most k− 1 which splits G into two components A and B such that G/A is cyclically k-edge-connected or the size of G/A is at most k.

Proof: Let C be a cyclic edge-cut of G whose size is at most k− 1, chosen

such that the order of a component B of G\ C is as small as possible. Let A be the other component of G\ C. We need to show that the graph G/A has no cyclic edge-cut of size at most k − 1. Assume the opposite and let C_A
be such an edge-cut. Let C
be the set of edges of G corresponding to those in C_A
. Clearly, C
is a cyclic edge-cut of G, |C
| ≤ k − 1 and one of the components of G\ C
has fewer vertices than B — a contradiction.

Consider a subgraph H of a graph G and an integer d≥ 1. A path P in G is a d-connecting path for H if

(i) the endvertices of P are in H,

(ii) no edge nor internal vertex of P is contained in H, (iii) the length of P is at most d.

The d-connector conn_d(H) of H is the smallest d-closed subgraph of G which contains H. A sequence (P₁, . . . , P_k) of paths in G is a construction sequence for conn_d(H) if, for each i = 0, . . . , k − 1, the path P_i+1 is d-connecting for

and H(k) _{= conn}

d(H). Clearly, there is at least one construction sequence for

conn_d(H). Although the construction sequence is not uniquely determined, the d-connector itself is.

We now show that if the girth of a graph G is sufficiently large, then the d-connector (as well as its neighborhood) of a subgraph with few edges is a forest of small size.

Lemma 14 Let d > ≥ 1. Suppose that a subgraph H of a graph G contains

at most  edges and no isolated vertices. If the girth of G is greater than (d + 1), then the neighborhood of the d-connector of H is a forest. Furthermore, the size of conn_d(H) is less than d and its maximum degree is at most .

Proof: Let (P₁, . . . , P_k) be a construction sequence for conn_d(H). Let c(H) denote the number of components of H. Observe that c(H) ≤ . We claim that the endvertices of each P_i+1 are in different components of

H(i) = H ∪ P₁∪ · · · ∪ P_i.

If not, then let i₀ be the smallest index i violating the claim. Clearly, H(i0+1)

contains a cycle. Furthermore, i₀ < . To see this, observe that the numbers of components of the graphs

H, H(1), H(2), . . . , H(i0)

form a decreasing sequence; since c(H)≤ , i₀ must be less than .

Now i₀ <  implies that the size of H(i0+1)_{, and hence also the length of}

the cycle it contains, is at most  + i₀d≤ (d + 1). This contradicts the fact that the girth of G is greater than (d + 1).

We have shown that each path P_i+1joins vertices in different components of H(i)_{. Since H itself contains no cycles (by the girth assumption), we}

infer that conn_d(H) is a forest. Its neighborhood must be a forest as well. This follows from the fact that no two vertices of conn_d(H) have a common neighbor outside conn_d(H) (as there is no 2-connecting path for conn_d(H)). It remains to prove the stated bounds on the size and the maximum degree of conn_d(H). Since k < c(H) (recall that k stands for the number of paths in the construction sequence), the size of conn_d(H) is at most  + d(− 1) < d. For the bound on the maximum degree, consider a vertex x∈ V (H). Let us say that the degree of x in H is s. Since the size of H is at most  and each of its components contains an edge, c(H) ≤  − s + 1 and hence k ≤  − s.

It follows that adding the paths P₁, . . . , P_k to H can increase the degree of x by no more than − s. Thus, the degree of x in conn_d(H) is at most . Since x was arbitrary, the maximum degree of conn_d(H) is at most  as claimed.

We are now ready to prove our main theorem:

Theorem 15 For any integer ∆ ≥ 1 and real ε > 0, there exists a positive

integer g such that if G is a graph of maximum degree ∆ and girth at least g, then

χ
_c(G)≤ ∆ + ε.

Proof: We may clearly assume that ∆ ≥ 3. Fix d ≥ ∆ and g ≥ (d + 1)∆

large enough for the statement of Theorem 12 to hold. The proof proceeds by induction on the size of G. The assertion of the theorem is trivial if |E(G)| ≤ ∆. Furthermore, we may assume that G is connected.

If G is cyclically ∆-edge-connected, then G has a ∆-decomposition by Proposition 7. Note that this includes the case when G is acyclic. By The-orem 12 (where we set H to be any subgraph consisting of a single edge), χ
_c(G)≤ ∆ + ε. Otherwise, since |E(G)| > ∆, Lemma 13 implies that there is a cyclic edge-cut F of size at most ∆− 1 whose removal splits G into components A and B such that G/A is cyclically ∆-edge-connected or the size of G/A is at most ∆. Let F
denote the d-connector of F in the union of B and F . Lemma 14, used with  = ∆− 1, implies that N(F
) is a forest. Since B contains a cycle, N (F
)\ F has fewer edges than B. Thus, if we set G_A to be the union of A with N (F
), the size of G_A is smaller than the size of G. At the same time, G_A has girth at least g and (by Lemma 14) its maximum degree is at most ∆. By induction (needed only if the maximum degree is precisely ∆), G_A has a (∆ + ε)-edge-coloring c_A.

The graph G/A has maximum degree at most ∆. We claim that it has a ∆-decomposition. As shown above, G/A is either cyclically ∆-edge-connected (in which case it has a ∆-decomposition by Proposition 7), or it has at most ∆ edges (in which case there is a trivial ∆-decomposition). Therefore, B∪ F is ∆-decomposable as well. Since F
is a d-closed subgraph of B∪F of size less than d∆, the precoloring of N(F
) induced by c_Aextends to a (∆ + ε)-coloring c_B of B ∪ F by Theorem 12. The combination of c_A and c_B is the desired (∆ + ε)-coloring of G.

An immediate corollary of Theorem 15 is the following:

Corollary 16 For any integer ∆ ≥ 1 and real ε > 0, there exists a positive

integer g such that if G is a graph of maximum degree ∆ and girth at least g, then

χ
_f(G)≤ ∆ + ε, where χ
_f denotes the fractional chromatic index.

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