- Email: [email protected]

Electronic Notes in Discrete Mathematics 44 (2013) 9–14 www.elsevier.com/locate/endm

b-chromatic index of graphs

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Carlos Vin´ıcius G.C. Lima a,1 N´ıcolas A. Martins a,1 Leonardo Sampaio b,1 Marcio C. Santos a,1 Ana Silva b,1 a

Departamento de Computa¸c˜ ao, Universidade Federal do Cear´ a(UFC), Brazil b

Departamento de Matem´ atica, UFC, Brazil

Abstract A b-coloring of the vertices of a graph is a proper coloring where each color class contains a vertex which is adjacent to a vertex in each other color class. The bchromatic number of G is the maximum integer χb (G) for which G has a b-coloring with χb (G) colors. This problem was introduced by Irving and Manlove in 1999, where they showed that computing χb (G) is N P-hard in general and polynomialtime solvable for trees. A natural question that arises is whether the edge version of this problem is also N P-hard or not. Here, we prove that computing the bchromatic index of a graph G is N P-hard, even if G is either a comparability graph or a Ck -free graph, and give some partial results on the complexity of the problem restricted to trees. Keywords: b-chromatic index, edge coloring, complexity, caterpillars, trees.

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Email addresses: [email protected] (Lima), [email protected] (Martins), [email protected] (Sampaio), [email protected] (Santos), [email protected] (Silva) 2 Partially supported by CAPES, FUNCAP and CNPq/Brazil. 1571-0653/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.endm.2013.10.003

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C.V.G.C. Lima et al. / Electronic Notes in Discrete Mathematics 44 (2013) 9–14

Introduction

Let G be a simple graph 3 and suppose that we have a proper coloring of G for which there exists a color class c such that every vertex v in c is not adjacent to any vertex in at least one other color class; then we can separatedly change the color of each vertex in c to obtain a proper coloring with fewer colors. This heuristic can be applied iteratively, but we cannot expect to reach the chromatic number of G, since the coloring problem is N P-hard. On the basis of this idea, Irving and Manlove introduced the notion of b-coloring in [14]. Intuitively, a b-coloring is a proper coloring that cannot be improved by the above heuristic, and the b-chromatic number χb (G) measures the worst possible such coloring. Finding χb (G) was proved to be N P-hard in general graphs [14], and remains so even when restricted to bipartite graphs [17] or to chordal graphs [12]. However, this problem is polynomial when restricted to some graph classes, including trees [14], cographs and P4 -sparse graphs [3], P4 -tidy graphs [20], cacti [6], some power graphs [8,9,10], Kneser graphs [11,15], some graphs with large girth [5,16,18], etc. Also, some other aspects of the problem were studied, as for example, the b-spectrum of a graph [1], and b-perfect graphs [13]. In this article, we propose to study a natural variation of the problem: coloring the edges of a graph under the same constraints (alternatively, to investigate the b-chromatic number of line graphs). In this context, we deﬁne an edge b-coloring of G as a proper edge-coloring c such that each color class contains at least one edge adjacent to edges in each other color class; such an edge is called a b-edge of c. A subset of b-edges containing exactly one b-edge of each color is called a basis of c. The b-chromatic index of G is thus deﬁned as the maximum integer χb (G) for which G has an edge b-coloring with χb (G) colors. In [14], Irving and Manlove also introduced a simple upper bound for χb (G), the m-degree of G, m(G), which we adapt to the edge b-coloring context. Given a graph G and an edge e ∈ E(G), let the degree d(e) of e be the number of edges that are incident with e. The m -degree of G is the maximum integer k for which there are at least k edges of degree at least k −1; we denote it by m (G). Clearly, χ (G) ≤ χb (G) ≤ m (G) Here, we prove that deciding if χb (G) equals m (G) is N P-complete. Since it is known that the b-chromatic number of a tree T is at least m(T ) − 1 [14], 3

The graph terminology used in this paper follows [2].

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an interesting question is whether or not the analog holds for the b-chromatic index of trees. However, the answer is no, as can be seen in [18] where the authors show that the diﬀerence m (G) − χb (G) for a tree G can be arbitrarily large. Nevertheless, we prove that this diﬀerence is at most 1 when G is a caterpillar. Also, we give some special cases where an edge b-coloring of a tree with k colors can be found in polynomial time, k given as input. For ﬁxed k, the problem can be solved in polynomial time for trees [12,19]. We recall that computing χb (G) when G is chordal is N P-hard and nothing is known about the b-chromatic number of subclasses of chordal graphs. The present study has roots on the investigation of the b-chromatic number of block graphs.

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N P-completeness

We consider the following problem and prove a reduction from d-edge-colorability of d-regular graphs, d ≥ 3, which is proved to be N P-complete in [4] even if G is either a comparability graph or a Ck -free graph. Edge b-Chromatic Problem Instance: A graph G. Question: Is χb (G) equal to m (G)? Theorem 2.1 The Edge b-Chromatic Problem is N P-complete, even if G is either a comparability graph or a Ck -free graph, for k ≥ 4. Sketch of proof: The problem is clearly in N P. The reduction is obtained as follows. Given a d-regular graph G, construct H from G by adding vertices {w, w , w1 , · · · , wd } and edges {ww } ∪ {w wi | i ∈ [1, d]} ∪ {wx | x ∈ V (G)}. Also, add n vertices adjacent to wi , for each i ∈ [1, d]. One can see that m (H) = n + d + 1 and it is possible to prove that H has an edge b-coloring with n+d+1 colors if and only if G has a d-edge-coloring. The construction is also closed under the graph classes Ck -free, k ≥ 4, and comparability graphs. Thus the theorem follows. 2

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Caterpillars

We call an edge e ∈ E(G) dense (k-dense) if d(e) ≥ m (G) − 1 (d(e) ≥ k − 1) and we represent the set of all dense edges (k-dense edges) of G by D (G) (Dk (G)). A caterpillar is a tree in which every vertex is in a central path or at distance 1 from this central path. Theorem 3.1 If G = (V, E) is a caterpillar, then χb (G) ≥ m (G) − 1.

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Sketch of proof: Let m denote m (G) and L(G) denote the line graph of G. We suppose that ω(L(G)) < m − 1, as otherwise the result clearly follows. Thus, every edge with degree at least m − 1 lies in the central path of G. Let E = {e1 , . . . , em −1 } be any subset of m − 1 edges of G with degree at least m − 1, numbered in the order they appear on the central path, from left to right. We ﬁrst color each ei with i, i ∈ {1, . . . , m − 1}. Then, for each i ∈ {1, . . . , m − 2}, we choose an edge e incident on the left extremity of ei which is not in the central path (this edge exists because m > ω(L(G)) + 1): if i + 2 < m and the right extremity of ei equals the left extremity of ei+1 , then color e with i + 2; otherwise, color e with i + 1. Because we can repeat at least one color in the neighborhood of each ei (recall that we are using m − 1 colors and that each ei ∈ E has degree at least m − 1), we can prove that this simple precoloring can be extended to an edge b-coloring of G with m − 1 colors. In fact, our proof holds as long as we can repeat at least one color in the neighborhood of each ei . Thus, we can actually ﬁnd an edge b-coloring of G with k colors, for all k ∈ {ω(L(G)) + 1, · · · , m − 1}. 2

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Partial results on trees

In this section, we investigate the b-chromatic index of trees and obtain some polynomial algorithms conditioned by the existence of certain subsets of kdense edges. In fact, the results presented here actually hold for vertex bcoloring of block graphs in general (it is known that the line graph of a tree is a claw-free block graph). We will thus present the results in terms of the traditional b-coloring of vertices. Consider a block graph G and a subset W ⊆ Dk (G) of cardinality k. In this section, we analyse the existence of a b-coloring with basis W . We ﬁrst need some additional deﬁnitions. A vertex u ∈ V (G) \ W is said to be encircled by W if every w ∈ W is either adjacent to u or has a common neighbor w ∈ W with u of degree k − 1. This concept was also introduced by Irving and Manlove and it is not hard to see that if W encircles some vertex, then it cannot be the basis of a b-coloring of G. In [14], they also proved that for trees this condition was also suﬃcient, which is not the case for block graphs. In Figure 1, for example, W does not encircle any vertex and cannot be a basis of a b-coloring. We prove that some special neighbors of W pose greater diﬃculty for b-coloring the graph. A link of W is a path between vertices of W of length 2 or 3 having intermediate vertices not in W . A vertex u ∈ V (G) \ W is a link vertex of W if there exists a link containing u. Also, if no link passing by u is an induced

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Fig. 1. One cannot color the black vertices so that W (grey vertices) can be a basis.

path, we say that u is a side vertex. For example, all the link vertices in Figure 1 are side vertices. Theorem 4.1 Let G be a block graph and W ⊆ Dk (G) be a subset of cardinality k that encircles no vertex, k ∈ {ω(G) + 1, · · · , m(G)}. If W has no side vertices, then there exists a b-coloring of G with k colors. The proof of Theorem 4.1 ﬁrst colors N [W ] using a technique that resembles the one used by Irving and Manlove in [14]. However, extending this partial coloring to a b-coloring of G is not a trivial task. The following lemma is applied and we emphasize that it holds for exteding any partial b-coloring of a block graph, no matter the structure of W . Its proof uses recoloring techniques, managing the loss of b-vertices. Lemma 4.2 Let G be a block graph, W ⊆ Dk (G) of cardinality k, k ∈ {ω(G) + 1, · · · , m(G)}, and ψ be a partial coloring that colors N [W ] and is such that each vertex of W is already a b-vertex of a distinct color. Then there exists a b-coloring of G with k colors.

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