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bColoring graphs with large girth
Journal of the Brazilian Computer Society volume 18, pages 375–378 (2012)
Abstract
A bcoloring of a graph is a coloring of its vertices such that every color class contains a vertex that has a neighbor in all other classes. The bchromatic number of a graph is the largest integer k such that the graph has a bcoloring with k colors. We show how to compute in polynomial time the bchromatic number of a graph of girth at least 9. This improves the seminal result of Irving and Manlove on trees.
1 Introduction
Let G be a simple graph. A proper coloring of G is an assignment of colors to the vertices of G such that no two adjacent vertices have the same color. The chromatic number of G is the minimum integer χ(G) such that G has a proper coloring with χ(G) colors. Suppose that we have a proper coloring of G and there exists a color h such that every vertex x with color h is not adjacent to at least one other color (which may depend on x); then we can change the color of these vertices and thus 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 hard. On the basis of this idea, Irving and Manlove introduced the notion of bcoloring in [15]. Intuitively, a bcoloring is a proper coloring that cannot be improved by the above heuristic, and the bchromatic number measures the worst possible such coloring. More formally, consider any vertex coloring of G. A vertex u is said to be a bvertex (for this coloring) if u has a neighbor colored with each color different from its own color. A bcoloring of G is a proper coloring of G such that each color class contains a bvertex. A basis of a bcoloring is a set of bvertices, one for each color class. The bchromatic number of G is the largest integer k such that G has a bcoloring with k colors. We denote it by χ_{ b }(G).
Naturally, a proper coloring of G with χ(G) colors is a bcoloring of G, since it cannot be improved. Hence, χ(G)≤χ_{ b }(G). For an upper bound, observe that if G has a bcoloring with k colors, then G has at least k vertices with degree at least k−1 (a basis of the bcoloring). Thus, if m(G) is the largest integer such that G has at least m(G) vertices with degree at least m(G)−1, we know that G cannot have a bcoloring with more than m(G) colors, i.e.,
This upper bound was introduced by Irving and Manlove in [15]. They showed that the difference between χ_{ b }(G) and m(G) can be arbitrarily large for general graphs. They proved that χ_{ b }(G) is equal to m(G) or m(G)−1 when G is a tree, and provided a polynomial time algorithm that computes χ_{ b }(G) for every tree. In addition, the problem was proved to be NPhard in general graphs [15], and remains so even when restricted to bipartite graphs [22]. These concepts have received much attention recently; for example, see [1–27].
Many of these works investigate the bchromatic number of graphs under assumptions that involve the existence of large cycles. For example, Irving and Manlove’s algorithm for trees can actually work on graphs with girth at least 11, as noticed by A. Silva in [26]. Also, there are a number of results about dregular graphs with girth at least 5 [3, 6, 18, 22, 23]. In this paper we improve Irving and Manlove’s result for graphs with large girth; more specifically, we prove the following.
Theorem 1
If G is a graph with girth at least 9, then χ _{ b }(G)≥m(G)−1.
Here is an outline of the proof of Theorem 1. A special set of vertices, called a good set of vertices, is defined and graphs are distinguished between having a good set and not having a good set. Next, we state some results by Irving and Manlove [15] and by Silva [26] that say that a graph G with girth(G)≥8 that does not have a good set cannot be bcolored with m(G) colors and has a bcoloring with m(G)−1 colors (hence, χ_{ b }(G)=m(G)−1); also, Silva proved that if G with girth at least 8 has a good set, then one can be found in polynomial time. Finally, and this is the original part of the paper, it is shown that if G with girth at least 9 has a good set, then χ_{ b }(G)=m(G). The proof of Theorem 1 yields a polynomial time algorithm that finds an optimal bcoloring of graphs with girth at least 9.
2 Definitions and partial results
In this section, we present some necessary definitions and the results by Irving and Manlove [15] and A. Silva [26] that complement our proof. The graph terminology used in this paper follows [4].
Let G be a simple graph. We denote by V(G) and E(G) the sets of vertices and edges of G, respectively. If X⊆V(G), then N^{X}(u) represents the set N(u)∩X. The girth ofG is the size of a shortest induced cycle of G.
Recall that m(G) is the largest integer k such that G has at least k vertices with degree at least k−1. We say that a vertex u∈V(G) is dense if d(u)≥m(G)−1; and we denote the set of dense vertices of G by M(G).
Let W be a subset of M(G), and let u be any vertex in V(G)∖W. If u is such that every vertex v∈W is either adjacent to u or has a common neighbor w∈W with u such that d(w)=m(G)−1, then it is said that Wencircles vertexu (or that u is encircled by W). A subset W of M(G) of size m(G) is a good set if (our definition is slightly different from the one given by Irving and Manlove):

(a)
W does not encircle any vertex, and

(b)
Every vertex x∈V(G)∖W with d(x)≥m(G) is adjacent to a vertex w∈W.
Theorem 2
[15]
LetGbe any graph andWbe a subset ofM(G) withm(G) vertices. IfWencircles some vertexv∈V(G)∖W, thenWis not a basis of a bcoloring withm(G) colors.
Theorem 3
[26]
IfGis a graph with girth at least 8, thenGdoes not have a good set if and only if M(G)=m(G) andM(G) encircles a vertex inV(G)∖M(G). Moreover, a good set ofG (if any exists) can be found in polynomial time.
A part of the proof of Theorem 1 consists of the following theorem:
Theorem 4
[26]
LetGbe a graph with girth at least 8. IfGhas no good set, thenχ_{ b }(G)=m(G)−1.
Now, all we need to prove is that if G does have a good set, then G can be bcolored with m(G) colors, which is done in the next section.
3 Coloring graphs with a good set
In this section we prove the second part of the main theorem, namely:
Theorem 5
LetGbe a graph with girth at least 9. IfGhas a good set, thenχ_{ b }(G)=m(G).
Let W={v_{1},…,v_{m(G)}} be a good set of G. Our aim is to construct a bcoloring of G with m(G) colors such that, for each i∈{1,…,m(G)}, vertex v_{ i } is a bvertex of color i. We start by assigning color i to v_{ i }, for each i∈{1,…,m(G)}. Next, we extend this partial coloring to the rest of the graph in several steps. Before explaining each step, we need to introduce some other terminology and notation.
A link is any path of length two or three whose extremities are in W and whose internal vertices are not in W. Any interior vertex of a link is called a link vertex. Let L be the set of all link vertices.
We first color G[W∪L] in a way not to repeat too many colors in N(w), for all w∈W, and at the end we extend the obtained partial coloring to a bcoloring of G with m(G) colors. Let G′=G[W∪L], L_{1} be the set of vertices of L that have at least one neighbor in L and L_{2} be the set of vertices in L that have at least two neighbors in W. The steps below are followed in order in such a way that we only move on to the next step when all the possible vertices are iterated.

1.
For each x∈L_{1}, let x′∈N^{L}(x). Since x′∈L, there must exist v_{ i }∈N^{W}(x′); color x with i;

2.
For each v_{ i }∈W, let . Also, let . If q>1, then use colors i_{1},…,i_{ q } to color the uncolored vertices in in a way that x_{ j } is not colored with i_{ j } (it suffices to make a derangement of those colors on the vertices);

3.
Let x∈L_{2} still uncolored be such that there exists v_{ i }∈N^{W}(x) that has some neighbor y∈L_{1}. Let c be the color of y; color x with c and recolor y with j, for any v_{ j }∈N^{W}(x)∖{v_{ i }};

4.
Finally, if x∈L_{2} is still uncolored, we know that N^{L}(v_{ i })={x}, for all v_{ i }∈N^{W}(x). Since N^{L}(x)=∅, we can color x with i, for any v_{ i } that is not adjacent to x and has no common neighbor with x in W of degree m(G)−1, which exists as x is not encircled by W.
Suppose that the algorithm above produces a partial coloring that colors every vertex in L in such a way that, at the end, each v_{ i }∈W has at least as many uncolored neighbors as missing colors in its neighborhood. Since L is colored, we know that the uncolored neighbors of W form a stable set. Thus, we can independently color N(v_{ i }) in such a way that v_{ i } sees every other color, for all v_{ i }∈W. By the definition of a good set, we know that if d(v)≥m(G), then v is already colored; hence, the partial coloring can be greedily transformed into a bcoloring with m(G) colors. Now, to prove that the algorithm works, we show that after these steps the obtained partial coloring ψ satisfies:
P1 ψ is proper; and
P2 the number of uncolored neighbors of v_{ i } is at least the number of missing colors in N(v_{ i }), for each v_{ i }∈W.
Proof of Theorem 5
First, we make some observations concerning the coloring procedure. Note that L_{1}∩L_{2} is not necessarily empty, but all vertices in this subset are colored in Step 1. However, a vertex x∈L_{1}∩L_{2} may play a role in Step 2 in the following way: if x∈N(v_{ i }) and there exists , then x′ may be colored with color j for some v_{ j }∈N^{W}(x)∖{v_{ i }}, while the color of x remains unchanged. Also, note that, in Step 3, since , we have y∈L_{1}∖L_{2}. Hence, N^{W}(y)={v_{ i }} and, consequently, the color of y cannot be changed again. Thus (*) the color of y is changed at most once, for every y∈L_{1}. Finally, if x receives color i in Step 1, 2 or 3, then one of the following holds (fact (iii) holds because of (*)):

(i)
x receives color i in Step 1 and there exists a path 〈x,x′,v_{ i }〉, for some x′∈L_{1}; or

(ii)
x receives color i in Step 2 and there exists a path 〈x,v_{ j },x′,v_{ i }〉, for some v_{ j }∈W and x′∈L_{2}; or

(iii)
x receives color i in Step 3 and there exists a path 〈x,v_{ j },y,y′,v_{ i }〉, for some v_{ j }∈W, y∈L_{1}∖L_{2} and y′∈L_{1}; or

(iv)
x is recolored with color i in Step 3 and there exists a path 〈x,v_{ j },x′,v_{ i }〉, for some v_{ j }∈W and x′∈L_{2}∖L_{1}.
We first prove that P1 holds after Step 3. Suppose that there exists an edge wz such that ψ(w)=ψ(z)=i. Since G has no cycle of length at most 7, the paths defined in (i)–(iv) are shortest paths. Therefore, vertex v_{ i } has no neighbor colored i and hence, w,z∈L. Also, as wz∈E(G), we have w,z∈L_{1} and they are colored in Step 1 and maybe recolored in Step 3. By (i) and (iv), there exist a w,v_{ i }path P_{ w } and a z,v_{ i }path P_{ z }, both of length at most 3. Note that either P_{ w }+P_{ z }+wz contains a cycle of length at most 7 or one of these paths consists of the edge wz followed by the other path. Because G has girth at least 9, the latter case occurs. We get as contradiction as this implies that at least one path is defined by (i) and, thus, vertex v_{ i } has a neighbor colored i.
Now, we prove that P2 also holds after Step 3. We actually prove that, after Step 3, no color is repeated in N(v_{ i }), for each v_{ i }∈W. Suppose there exist a vertex v_{ j }∈W and w,z∈N(v_{ j }) such that ψ(w)=ψ(z)=i. First, consider the case v_{ i }∈{w,z}. Since the paths defined by (i)–(iv) are shortest paths, we see that (i) occurs for the vertex in {w,z}∖{v_{ i }}. We get a contradiction as this implies G has a cycle of length 4. Therefore we may assume v_{ i }∉{w,z}.
Now, by (i)–(iv), there exist a w,v_{ i }path P_{ w } and a z,v_{ i }path P_{ z }. Let ℓ_{ w } and ℓ_{ z } be the length of P_{ w } and P_{ z }, respectively. Clearly ℓ_{ w },ℓ_{ z }≤4. Note that either P_{ w }+P_{ z }+〈w,v_{ j },z〉 contains a cycle of length at most ℓ_{ w }+ℓ_{ z }+2 or either P_{ w } or P_{ z } consists of the path 〈w,v_{ j },z〉 followed by the other path. Since both w and z are at distance at least 2 from v_{ i } and ℓ_{ w },ℓ_{ z }≤4, the latter can only occur if one of the paths is defined by (i), say P_{ w }, and the other is defined by (iii), say P_{ z }. We get a contradiction as P_{ z }=〈z,v_{ j },w,y,v_{ i }〉 implies w is recolored in Step 3 and therefore, P_{ w } must be defined by (iv). Now, suppose that the former occurs, i.e., P_{ w }+P_{ z }+〈w,v_{ j },z〉 contains a cycle of length at most ℓ_{ w }+ℓ_{ z }+2. Because G has girth at least 9, we have ℓ_{ w }+ℓ_{ z }≥7. This implies that at least one of P_{ w } and P_{ z }, say P_{ z }, is defined by (iii), and the other is not defined by (i). Therefore z is colored in Step 3 and . Furthermore, w∈L_{1}∖L_{2} and N^{W}(w)={v_{ j }}. Therefore, since (i) does not occur for w, we find that P_{ w } must be defined by (iv). Thus the only choice for P_{ w } is 〈w,v_{ j },z,v_{ i }〉, a contradiction as P1 holds.
Finally, consider x to be colored during Step 4 with color i. By the choice of i we know that v_{ i }∉N(x). Thus, since N^{L}(x)=∅, Property P1 holds. Now, suppose that some v_{ j }∈N(x) is such that color i already appears in N(v_{ j }). Since N^{L}(v_{ j })={x} we must have v_{ i }∈N(v_{ j }) and, by the choice of i, d(v_{ j })>m(G)−1. Property P2 thus follows as i is the only repeated color in the neighborhood of v_{ j }. □
4 Conclusion
We showed that if G is a graph with girth at least 9, then χ_{ b }(G)≥m(G)−1, improving the result by Irving and Manlove [15]. We also give an algorithm that finds the bchromatic number of G in polynomial time.
In [25], Maffray and Silva conjecture that any graph G with no K_{2,3} as subgraph has bchromatic number at least m(G)−1. Observe that these graphs contain all graphs with girth at least 9; thus, we have given a partial answer to their conjecture. Actually, if their conjecture holds, then χ_{ b }≥m(G)−1 holds for every G with girth at least 5. However, a different approach is needed as our proof strongly relies on the fact that girth(G)≥9. Moreover, Theorem 5 does not hold for an infinite family of cacti with girth 5, as can be seen in [7]. This means that the situation where G has no good set is not the only situation where a graph G with girth at least 5 cannot be bcolored with m(G) colors.
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Acknowledgements
This work has been partially supported by Funcap (Fundação Cearense de Apoio ao Desenvolvimento Científico e Tecnológico) and CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico).
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Campos, V., de Farias, V.A.E. & Silva, A. bColoring graphs with large girth. J Braz Comput Soc 18, 375–378 (2012). https://doi.org/10.1007/s1317301200639
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DOI: https://doi.org/10.1007/s1317301200639