Co-k-plex vertex partitions

Transcription

1 Co-k-plex vertex partitions Benjamin McClosky John D. Arellano Illya V. Hicks November 7, 2012 Abstract This paper studies co-k-plex vertex partitions and more specifically co-2-plex vertex partitions. Co-k-plexes and k-plexes were first introduced in 1978 in the context of social network analysis. However, the study of co-k-plex vertex partitions, or the decomposition of graphs into degree bounded subgraphs, dates back at least to the work of Lovasz in In this paper, we derive analogues for well-known results on the chromatic number and present two algorithms for constructing co-2-plex vertex partitions. The first algorithm minimizes the number of partition classes. The second algorithm minimizes a weighted sum of the partition classes, where the weight of a partition class depends on the level of adjacency among its vertices. 1 Introduction A coloring partitions the vertex set of a graph G = (V, E) into subsets of pairwise nonadjacent vertices. A classic problem in combinatorial optimization is to find a coloring that uses the smallest possible number of color classes. The minimum number of color classes required is known as the chromatic number χ(g). If V represents a set of objects and E the set of conflicting pairs, graph coloring solves the problem of dividing V into the minimum number of conflict-free subgroups. A second application of graph coloring arises from its relation to another classic problem in combinatorial optimization. The maximum clique problem asks for the largest subset of pairwise adjacent vertices in a graph. Since the color classes of a coloring are edgeless, a subset of pairwise adjacent vertices meets each color class at most once. Consequently, χ(g) is an upper bound on the cardinality of a maximum clique, and researchers use graph coloring in branch and bound solvers for the maximum clique problem [2, 24, 26]. Generalized graph coloring describes the partitioning of vertices into classes whose induced subgraphs satisfy particular constraints [25]. For example, k-improper colorings have the property that each color class induces a subgraph of maximum degree at most k [1, 9, 14]. The generalization to k-improper colorings suggests two optimization problems. The first seeks to partition a graph into degree-bounded subgraphs using the smallest possible number of partition classes. This solves the problem of dividing V into the minimum number of subgroups such that each vertex has a bounded number of conflicts within its partition class. The k-improper chromatic number is associated with this problem and has been studied in a variety of contexts [4, 8, 13, 15]. Much of this research focuses on random 1

2 graphs and generalizations of the Four Color Theorem. Some applications of k-improper coloring include radio-frequency assignment [13] and network security [22]. The second problem minimizes a weighted sum of the partition classes. In this paper, the weight of a partition class will depend on the level of adjacency among its vertices. Such a weighted partition produces a bound on the cardinality of subgraphs defined as degreebased clique relaxations [21]. This optimization problem appears to be new, and the main contribution of this paper is to examine its properties and provide the first exact algorithm for the case k = 2. This paper is organized as follows. Section 2 discusses some relevant definitions and notation. Section 3 explores the relationship between degree-based clique relaxations and the weighted version of degree-bounded vertex partitions. Section 4.1 offers a straightforward adaptation of a well-known graph coloring algorithm [17] to solve the problem of minimizing the number of partition sets. Section 4.2 uses the algorithm from Section 4.1 to design the first exact algorithm for the weighted degree-bounded vertex coloring problem when k = 2. Section 6 summarizes and suggests some future research directions. 2 Preliminaries All graphs G = (V, E) in this paper are finite, undirected, and simple. Given a vertex v V, define N G (v) := {u V uv E}, deg G (v) := N G (v), (G) := max v V deg G (v), and δ(g) := min v V deg G (v). Let G[K] denote the subgraph induced by K V. In this paper, k 1 is always a positive integer. Definition 1. K V induces a k-plex if δ(g[k]) K k. Definition 2. C V induces a co-k-plex if (G[C]) k 1. Definition 3. A partition of the vertex set into disjoint, nonempty co-k-plexes defines a co-k-plex coloring of G. Definitions 1 and 2 are due to Seidman and Foster [23]. This paper uses the terms cok-plex colorings and co-k-plex partitions interchangeably. Co-k-plexes are also known as (k-1)-dependent or (k-1)-stable sets [13]. Notice that 1-plexes and co-1-plexes are cliques and stable sets, respectively. Let ω k (G) denote the cardinality of a largest k-plex in G, α k (G) the cardinality of a largest co-k-plex in G, and Π the set of all co-k-plex colorings of G. Definition 4. The weighted co-k-plex chromatic number of G is defined as χ k (G) := min{ C P ω k (G[C]) : P Π}. Definition 5. The co-k-plex chromatic number of G is defined as χ k (G) := min{ P : P Π}. 2

3 χ k (G) is exactly the (k-1)-improper chromatic number [25]. A χ k -optimal coloring partitions V using the smallest possible number of co-k-plex sets. A χ k -optimal coloring C 1,..., C m satisfies χ k (G) = m i=1 ω k(g[c i ]) and thus χ k (G) m m ω k (G[C i ]) = χ k (G). i=1 Notice also that χ 1 (G) = χ 1 (G) = χ(g). Moreover, a coloring is χ 1 -optimal if and only if it is χ 1 -optimal. However, this relationship fails for k > 1. To see this, consider the trivial example of k pairwise non-adjacent vertices. The unique χ k -optimal coloring consists of a single color class. On the other hand, assigning each vertex to a distinct color class defines a χ k -optimal coloring which uses k color classes. It turns out that one can still derive a bound relating χ k (G) and χ(g). Theorem 1. For any graph G, χ k (G) kχ(g). Proof. For this result, we will use induction on χ(g). Since the base case is trivial, we can assume without loss of generality that χ(g) > 1. Let C G be an optimal family of color classes for G, and consider the stable set A C G. Now by induction, χ k (G \ A) kχ(g \ A). Now, consider CG\A k to be an optimal family of weighted co-k-plex color classes for G \ A. Notice that CG\A k A forms a co-k-plex coloring of G and χ k(g[a]) k. Hence, χ k (G) χ k (G \ A) + k kχ(g \ A) + k kχ(g). Given the aforementioned example where an optimal χ k (G)-coloring can have more color classes than an optimal χ(g)-coloring, we may also want to restrict our χ k (G)-coloring to the smallest number of color classes possible. A co-k-plex C is deficient whenever C < k. A deficient co-k-plex C satisfies ω k (G[C]) = C. Define a compact co-k-plex coloring as a co-k-plex coloring that has at most one deficient co-k-plex set. Lemma 1. Every co-k-plex coloring C 1,..., C m can be changed into a compact co-k-plex coloring C 1,..., C p such that p m and m i=1 ω k(g[c i ]) = p i=1 ω k(g[c i]). Proof. Consider the co-k-plex coloring C 1,..., C m. Suppose there are two deficient co-k-plexes C i and C j. It follows that ω k (G[C i ]) + ω k (G[C j ]) = C i + C j. Choose a vertex v C j. Define C j := C j \ {v} and C i := C i {v}. Now C i {v} k ensures that C i and C j both remain co-k-plexes. Moreover, ω k (G[C i]) + ω k (G[C j]) = ( C i + 1) + ( C j 1) = ω k (G[C i ]) + ω k (G[C j ]). Continue moving vertices from C j to C i until either C j = or C i = k, in which case the number of deficient sets has been reduced. This procedure can be repeated until the co-k-plex coloring C 1,..., C p is compact. It is also clear that p m since the procedure can only reduce the number of partition sets in the co-k-plex coloring. We will use this lemma to verify a nice relationship between weighted co-2-plex colorings and unweighted co-2-plex colorings. McClosky and Hicks [21] use weighted co-k-plex coloring 3

4 to produce an upper bound on ω k (G). Let S 1,..., S m be a co-k-plex coloring of G, and let K V be a maximum k-plex in G. Observe that ω k (G) = K = m K S i i=1 m ω k (G[S i ]), i=1 where the inequality follows from the fact that k-plexes are closed under set inclusion [23]. Now Definition 4 implies that χ k (G) ω k (G). The following section explores this relationship in more detail. 3 The Relationship between χ k (G) and ω k (G) This section analyzes the relationship between χ k (G) and ω k (G). In particular, we address the following two questions. Can we characterize a group of graphs that satisfy χ k (G) = ω k (G), and can we bound the gap between the two invariants? Attempts to address these questions are given in Sections 3.1 and 3.2, respectively. 3.1 The k-plex-matroid Property This section analyzes the relationship between χ k (G), ω k (G) and independence systems derived from graphs with respect to k-plexes. In particular, we examine the relationship when these corresponding independence systems are in fact matroids. Recall that a finite set X and a family I of subsets of X define a matroid if the following axioms hold: 1. I 2. I I I implies I I 3. For any subset A X, every maximal independent subset in A has the same cardinality Also, if (X, I) satisfies the first two axioms then (X, I) is an independence system. Now, given a graph G = (V, E), define K = {K V : δ(g[k]) K k}. Notice that K is the set of k-plexes in G and (V, K) defines an independence system. Definition 6. The graph G is k-plex matroidal if (V, K) defines a matroid. Theorem 2. If G is k-plex matroidal, then ω k (G ) = χ k (G ) for all vertex-induced subgraphs G G. Proof. Let M := (V, K) define a matroid. Given any vertex-induced subgraph G = (V, E ), define D := V \ V and K = {K V : δ(g[k]) K k}. Observe that (V, K ) = (V \ D, K ) =: M \ D 4

5 is again a matroid known as a deletion matroid, so it suffices to show χ k (G) = ω k (G). Define x(a) = a A x a, S = {S V : (G[S]) k 1}, and S v = {S S : v S}. Consider the following dual pair of linear programs: max{x(v ) : x 0, x(s) ω k (G[S]) for all S S} (1) min{ S S ω k (G[S])y S : y 0, y(s v ) 1 for all v V }. (2) Since M is a matroid, theorems (19) and (21) of Edmonds [11] imply that the optimal solutions for (1) and (2) are integral. Observe that ω k (G) and χ k (G) are the optimal objective values for (1) and (2), respectively. Moreover, ω k (G) = χ k (G) by strong duality. Notice that the converse of Theorem 2 is not true. Indeed, one can easily find a co-k-plex where k > 1 that does not satisfy the k-plex matroidal property. For example, consider the cycle on eight nodes for k = 3 which has a maximal 3-plex of size 3 and a maximum 3-plex of size 4. In contrast, every vertex induced subgraph of a co-k-plex satisfies the bound with equality. Next, we do show that k-plexes satisfy the k-plex matroidal property. Corollary 1. If G is a k-plex, then G is k-plex matroidal. Proof. Given any K V and v V \ K, K {v} defines a k-plex. It follows that all maximal k-plexes have cardinality ω k (G) = V. Recall that an r-partite graph is r-colorable. The complete r-partite graphs have all possible edges between distinct color classes. Next, we show that k plexes are not the only type of graphs that satisfy the k-plex matroidal property. Hence, the class of k-plex matroidal graphs is not just strictly k-plexes. Theorem 3. If G is the complete r-partite graph K n1,...,n r, then G is k-plex matroidal. Proof. The proof will show that all maximal k-plexes in G have the same cardinality. Let K be a maximal k-plex in G and S i the i th partition class. Clearly, K S i S i = n i. In addition, K S i k. For if not, let v K S i, and notice that N G (v) S i = implies deg G[K] (v) = K K S i < K k, which contradicts that K is a k-plex. Therefore, K S i min{k, n i } for each S i. Suppose for contradiction that K = r i=1 K S i < r i=1 min{k, n i}. Then there exists a j such that K S j < min{k, n j }, and K S j < n j implies that there exists a vertex v S j \ K. Consider the set K := K {v} and a vertex u K \ S j. Since uv E, deg G[K ](u) = deg G[K] (u) + 1 ( K k) + 1 = K k. Now suppose u K S j. Observe that deg G[K ](u) = deg G[K] (u) = K K S j > K k since uv E and K S j < k. It follows that deg G[K ](u) K k + 1 = K k. Thus, since deg G[K ](u) = deg G[K ](v), K is a k-plex in G, which contradicts the maximality of K. It follows that all maximal k-plexes in G have cardinality r i=1 min{k, n i}, so G is k-plex matroidal. 5

6 3.2 A theorem of Erdős In 1959, Erdős [12] showed that the difference χ 1 (G) ω 1 (G) can be arbitrarily large. More precisely, he showed that for every integer r 1, there exists a graph G with girth g(g ) > r and chromatic number χ(g ) > r. Observe that g(g) > 3 implies ω 1 (G) 2. Therefore, the theorem establishes the existence of graphs with high chromatic number and low clique number. Analogously, one might ask if the gap between χ k (G) and ω k (G) can also become arbitrarily large. This section uses the Erdős theorem to show that χ k (G) ω k (G) can be arbitrarily large. The proofs have been adapted from [10]. Let G n,p be the random graph on n vertices where each edge exists with probability 0 p 1. Let q = 1 p. Lemma 2. Every co-k-plex S has at most S (k 1) edges. 2 Proof. E(S) = 1 2 v V (S) deg G[S](v) 1 S (k 1) 2 v V (S)(k 1) =, where the inequality 2 follows from the definition of co-k-plex. Lemma 3. For all integers n t k + 1, the probability that G G n,p has a co-k-plex of size t is at most ( ) n P [α k (G) t] q t(t k)/2. t Proof. Consider a fixed t-set U V. By Lemma 2, the event that U is a co-k-plex is contained in the event that t(k 1) E(G[U]). (3) 2 Thus, the probability of the latter is an upper bound on the probability of the former. Now (3) requires that at least ( ) t 2 t(k 1) edges are missing. That is, (3) occurs with probability 2 at most q (t 2) t(k 1) 2. The lemma follows from the fact that G contains ( n t) t-sets U. It is worth mentioning that if t min{k, n}, then every t-set is a co-k-plex, and Lemma 3 fails. Theorem 4. Given any integer r > k, there exists a graph G with girth g(g) > r and co-k-plex chromatic number χ k (G) > r. Proof. For n large, suppose t n 2r > k and (8r ln n)n 1 p 1. By Lemma 3, Therefore, P [α k t] Now lim n n 1 e k/2 = 0, so ( ) n q t(t k)/2 n t q t(t k)/2 = (nq (t k)/2 ) t (ne p(t k)/2 ) t. t P [α k t] (ne pt/2 e pk/2 ) t (ne 2(ln n) e k/2 ) t = (n 1 e k/2 ) t. P [α k n 2r ] < 1 2 (4) for sufficiently large n. 6

7 On the other hand, fix ɛ with 0 < ɛ < 1/r, and let X(G) denote the number of cycles of length at most r in G G n,p. Erdős showed (see [10]) that for large n and p = n ɛ 1, P [X n 2 ] < 1 2. (5) Finally, fix n large enough to satisfy (4), (5), and n ɛ 1 (8r ln n)n 1. Let p = n ɛ 1. There exists a G G n,p such that α k (G) < n and G has less than n cycles of length at 2r 2 most r. Construct the graph H by removing a vertex from each cycle of length at most r. Then H n and g(h) > r. Furthermore, α 2 k(h) α k (G) < n implies that any co-k-plex 2r coloring of H requires more than r co-k-plex sets. Consequently, χ k (H) > r. Corollary 2. Given any integer r > k + 2, there exists a graph G with χ k (G) > r and ω k (G) < k + 2. Proof. If ω k (G) k + 2, then G contains a k-plex K of cardinality k + 2. Moreover, δ(g[k]) 2 by definition of k-plex. It follows that G[K] G contains a cycle of length at most k + 2 = K. Therefore, g(g) > k + 2 implies that ω k (G) < k + 2. The assertion now follows from Theorem 4. 4 Algorithms This section develops algorithms for finding degree-bounded vertex partitions. Section 4.1 contains an exact χ k -coloring algorithm. Section 4.2 shows how to find χ 2 -optimal colorings using the χ 2 -coloring algorithm. 4.1 χ k -optimal coloring In [17], Kubale and Jackowski present a generalized implicit enumeration algorithm for graph coloring which subsumes a number of previous combinatorial approaches [5, 6, 7, 16]. This section adapts the Kubale and Jackowski algorithm to find χ k -optimal colorings. Figure 1 contains the generalized implicit enumeration algorithm as given in [17]. Before running the algorithm, the vertex set of G is ordered (v 1,..., v n ) such that deg G (v i ) deg G (v i+1 ). The vertex ordering can either remain static or change dynamically throughout the algorithm. The array C stores a partial co-k-plex coloring. The array C stores the incumbent co-k-plex coloring. For each 1 i n, F C(i) stores the set of feasible colors for v i with respect to the current partial coloring C. In other words, F C(i) consists of partition classes S such that S {v i } is a co-k-plex. CP is the set of current predecessors. These vertices are the candidates for backtracking. The main difference between traditional graph coloring and χ k -coloring is the structure of the partition classes, so adapting the coloring algorithm in Figure 1 amounts to finding an appropriate definition for the set of feasible colors F C(i). Given the partial co-k-plex coloring S 1,..., S r, define P (i) = {j : S j {v i } is not a co-k-plex} {ub}. The set of feasible colors is defined as F C(i) = {1, 2,..., max j<i C (j) + 1} P (i). This definition forces each partition class to be a co-k-plex. 7

8 function implicitenum(g, n) 1. ub = n + 1; r = 1 2. loop 3. FORWARDS(r) 4. BACKWARDS(r) 5. if r = 0 then break 6. repeat end function FORWARDS(r) 7. for i = r to n 8. reorder uncolored vertices v i,..., v n 9. if r = 1 or r < i then determine F C(i) 10. if F C(i) = then r = i; return 11. C (i) = min(f C(i)) 12. repeat 13. C = C ; ub = max(c) 14. r = least i such that C(i) = ub end function BACKWARDS(r) 15. CP = {1,..., r 1} 16. while CP 17. i = max(cp ); CP = CP {i} 18. F C(i) = F C(i) {C (i)} 19. if F C(i) then r = i; return 20. repeat 21. r = 0 end Figure 1: A generalized implicit enumeration algorithm for graph coloring [17]. 8

9 4.2 χ 2 -optimal coloring Section 4.1 shows how traditional graph coloring algorithms can solve the χ k -coloring problem. However, these algorithms do not apply directly to χ k -coloring since a χ k -coloring algorithm must consider the weight ω k (G[S i ]) of each partition class in a co-k-plex coloring S 1,..., S m.. Although a result of Balasundaram et al. [3] shows that the maximum size of a k-plex within a co-k-plex is 2k 2 + (k mod 2), no practical algorithm is known for this problem for k > 2. Here, we focus on the k = 2 case. This section focuses on the χ 2 -coloring problem. The proposed algorithm effectively reduces χ 2 -coloring to χ 2 -coloring. Lemma 1 implies that the set of compact co-k-plex colorings always contains a χ k -optimal coloring. As a result, an algorithm can restrict the search for a χ k -optimal solution by considering only compact co-k-plex colorings. For k = 2, Lemma 1 has an even stronger consequence. Recall that ω 2 (G[C]) = min{2, C } {1, 2} for any nonempty co-2-plex C [20]. Let Π min be the set of compact χ 2 -optimal colorings. Define C 1,..., C m as follows: If possible, choose a co-2-plex coloring in Π min with a deficient set, and let C m be that deficient set. If no such co-2-plex coloring exists, choose an arbitrary co-2-plex coloring in Π min. Lemma 4. C1,..., C m is a χ 2 -optimal coloring. Proof. Suppose for contradiction that C 1,..., C m is not χ 2 -optimal. C1,..., C m is compact, so C i 2 for i = 1,..., m 1. Therefore, m i=1 ω 2(G[ C i ]) = 2(m 1) + min{2, C m }. Now consider a χ 2 -optimal coloring C 1,..., C r. By Lemma 1, assume that C 1,..., C r is compact. Therefore, χ 2 (G) = r i=1 ω 2(G[C i ]) = 2(r 1) + min{2, C r }, where C r is the deficient set if one exists in C 1,..., C r. Finally, observe that which implies 2(m 1) + min{2, C m } > χ 2 (G) = 2(r 1) + min{2, C r }, min{2, C m } min{2, C r } > 2(r m). Case 1: Suppose r = m. It follows that min{2, C m } min{2, C r } > 0. Thus min{2, C m } = 2 and min{2, C r } = 1. In other words, Cm is not deficient; C r is deficient; and C 1,..., C r belongs to Π min. This contradicts the choice of C 1,..., C m. Case 2: Suppose r > m. Then min{2, C m } min{2, C r } > 2, a contradiction since min{2, C m } {1, 2} and min{2, C r } {1, 2}. Lemma 4 reduces the problem of finding a χ 2 -optimal coloring to that of finding an element in Π min. This is desirable as the algorithm from Section 4.1 can solve the latter problem. The proposed algorithm consists of two steps, both of which make use of implicit enumeration. The first step constructs a compact χ 2 -optimal coloring. The second step searches for a compact χ 2 -optimal coloring with one deficient set. If such a co-2-plex coloring exists, it is χ 2 -optimal by Lemma 4. If no such co-2-plex coloring exists, then the co-2-plex coloring from the first step is χ 2 -optimal by Lemma 4. The first step uses the coloring algorithm exactly as described in Section 4.1. In the second step, for each v V, the algorithm uses implicit enumeration to search for a χ 2 (G) 1 9

10 coloring of G v. If step two manages to find such a coloring in G v, then v is a deficient co-2-plex in a χ 2 -optimal coloring of G. 5 Computational Results Let A1 denote the Kubale-Jackowski inspired algorithm described in Section 4.1. Preliminary tests showed that dynamic vertex ordering always outperformed the static ordering, so we used dynamic ordering for the computational experiments. In determining the order, the algorithm always colored the vertex v i that maximized P (i). Ties were broken by choosing the vertex of larger degree. This dynamic reordering is analogous to DSATUR [5]. In order to investigate the effectiveness of A1, we compared the algorithm to an integer programming formulation of χ k (G)-coloring for k values 2, 3, and 4. The integer programming formulation is given in Figure 2 where κ is defined as (G)+1, a generalization of the k Brooks Theorem bound given by Lovasz [18]. The formulation was solved by using Gurobi. All computations were conducted on a Sun Ultra 24 Workstation (3.0Ghz). Both algorithms were allowed at most 1,000 seconds to converge to an optimal solution. Tables 1-5 compare the two algorithms for finding optimal χ k -colorings over a broad range of test instances. Specifically, the algorithms were tested on a set of random graphs and a subset of the DIMACS coloring instances. The random graph GN-P -i had N vertices and edge probability P. The sec column contains > 1000 whenever the corresponding 100 algorithm failed to converge in 1,000 seconds. In these cases, we reported the incumbent feasible solution, which offers an upper-bound on the optimal solution. Several interesting trends emerge as a result of our computational experiments. First, edge density impacted the performance of both algorithms. The random graphs clearly illustrated the importance of edge density; both algorithms performed well as P approached 10 and struggled as P approached 90. Second, the computational burden increased with the number of vertices. With the exception of the sparse graphs jean and anna, both algorithms failed on all DIMACS instances having more than 50 vertices. Similarly, both algorithms failed when N 60 and P 50 on the random graphs. Third, the value of k was not highly predictive of instance difficulty. Comparing the two methods, A1 appeared to generally outperform the IP. Notice that the IP did very quickly produce an optimal solution to games120 for k = 4, an instance where A1 failed to converge. Overall, however, there were only a few cases where the integer programming approach outperformed A1. For the small sample of DIMACS graphs, A1 had more instances finished within the time limit and provided either an optimal value or a better upper bound than the integer programming approach. Altogether, this evidence suggests that A1 offers an attractive alternative to the integer programming approach. Moreover, we conclude that this problem is tractable on sparse graphs. 10

11 ( N(v) k + 1)x v,c + min κ c=1 y c s.t. x v,c y c v V, c {1,... κ} (6) k x v,c = 1 v V (7) c=1 w N(v) x w,c N(v) v V, c {1,... κ} (8) x v,c, y c {0, 1} v V, c {1,... κ} (9) Figure 2: An integer programming formulation for the optimal χ k -coloring 11

12 Table 1: χ k -coloring χ 2 (G) χ 3 (G) χ 4 (G) G V E A1 sec IP sec A1 sec IP sec A1 sec IP sec myciel myciel myciel myciel > > jean > anna queen queen > queen > > > > > > 1000 homer > > > > > 1000 games > > > > > upper bound Table 2: χ k -coloring χ 2 (G) χ 3 (G) χ 4 (G) G V E A1 sec IP sec A1 sec IP sec A1 sec IP sec G G G G G G G G G G G G G G G G G G G G G > G > G > G > G > upper bound 12

13 Table 3: χ k -coloring χ 2 (G) χ 3 (G) χ 4 (G) G V E A1 sec IP sec A1 sec IP sec A1 sec IP sec G G G G G G G G G G G > G > G > > > 1000 G > > G > G > > > > > 1000 G > > > > > 1000 G > > > > > 1000 G > > > > > > 1000 G > > > > > 1000 G > > > > > > 1000 G > > > > > > 1000 G > > > > > > 1000 G > > > > > > 1000 G > > > > > > 1000 upper bound Table 4: χ k -coloring χ 2 (G) χ 3 (G) χ 4 (G) G V E A1 sec IP sec A1 sec IP sec A1 sec IP sec G G G G G G > > > > G > > > > > 1000 G > > > > G > > > G > > > > > 1000 G > > > > > > 1000 G > > > > > > 1000 G > > > > > > 1000 G > > > > > > 1000 G > > > > > > 1000 G > > > > > > 1000 G > > > > > > 1000 G > > > > > > 1000 G > > > > > > 1000 G > > > > > > 1000 G > > > > > > 1000 G > > > > > > 1000 G > > > > > > 1000 G > > > > > > 1000 G > > > > > > 1000 upper bound 13

14 Table 5: χ k -coloring χ 2 (G) χ 3 (G) χ 4 (G) G V E A1 sec IP sec A1 sec IP sec A1 sec IP sec G G G G G G > > > > > > 1000 G > > > > > > 1000 G > > > > > > 1000 G > > > > > 1000 G > > > > > > 1000 G > > > > > > 1000 G > > > > > > 1000 G > > > > > > 1000 G > > > > > > 1000 G > > > > > > 1000 G > > > > > > 1000 G > > > > > > 1000 G > > > > > > 1000 G > > > > > > 1000 G > > > > > > 1000 G > > > > > > 1000 G > > > > > > 1000 G > > > > > > 1000 G > > > > > > 1000 G > > > > > > 1000 upper bound 14

15 Table 6: χ 2 -coloring G V E χ 2 (G) sec G V E χ 2 (G) sec G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G > 1000 G G > 1000 G G > 1000 G G > 1000 G G > 1000 upper bound Table 7: χ 2 -coloring G V E χ 2 (G) sec G V E χ 2 (G) sec G G G G G G G G G G G G > 1000 G G > 1000 G G > 1000 G G > 1000 G G > 1000 G G > 1000 G > 1000 G > 1000 G > 1000 G > 1000 G > 1000 G > 1000 G G > 1000 G > 1000 G > 1000 G > 1000 G > 1000 G > 1000 G > 1000 G > 1000 G > 1000 G > 1000 G > 1000 G > 1000 G > 1000 G > 1000 G > 1000 G > 1000 G > 1000 G > 1000 G > 1000 G > 1000 G > 1000 upper bound 15

16 Table 8: χ 2 -coloring G V E χ 2 (G) sec myciel myciel myciel myciel * > 1000 jean * > 1000 anna queen queen queen homer * > 1000 games * > 1000 upper bound 16

17 We next evaluated the performance of the weighted co-2-plex coloring algorithm. Tables 6-8 contain computational results obtained by running the algorithm on the instances which were solved to χ 2 -optimality by A1. Observe that edge-density and vertex number had the same impact on algorithm performance as in the case of χ k -coloring. Also notice that most of these graphs satisfy χ 2 (G) = 2 χ 2 (G). These are exactly the graphs where step two of the algorithm failed to find a coloring with a deficient co-2-plex. One instance where this relationship might not be the case is for the test instance homer with an odd upper bound. It may also be true for other test instances were the optimal solution is unknown. 6 Conclusion This paper studies degree-bounded vertex partitions. In Section 3, we studied the relationship between χ k -optimal colorings and maximum k-plexes and the bound between them, ω k χ k. In this analysis, we discovered examples of graphs that satisfy a matroidal property with respect to k-plexes, which implies equality of the aforementioned bound for all node induced subgraphs of these types of graphs. Further analysis is warranted to discover other classes where the bound is also tight. We also derived algorithms for constructing degree-bounded vertex partitions. Section 4.1 offers a straightforward generalization of a traditional graph coloring algorithm. The resulting algorithm partitions the vertex set into the minimum number of co-k-plexes. This algorithm outperformed an integer programming approach to the same problem for most of the test instances given in Section 5. Section 4.2 shows how to find χ 2 -optimal colorings by reducing the problem to that of finding χ 2 -optimal colorings. This is the first known algorithm for constructing χ 2 -optimal colorings. Lemma 4 represents a key step in the reduction. An interesting open problem is to determine if χ k -coloring can be used to solve χ k -coloring problems for k > 2. Another avenue for future research is a polyhedral analysis of the co-k-plex coloring polytope. 7 Acknowledgements The authors are grateful for the National Science Foundation grant CMMI for partially supporting this research. References [1] J. A. Andrews and M. S. Jacobson. On a generalization of chromatic number. Proceedings of the sixteenth Southeastern international conference on combinatorics, graph theory, and computing, 47 (1985), pp [2] E. Balas and J. Xue. Weighted and unweighted maximum clique algorithms with upper bounds from fractional coloring. Algorithmica 15 (1996), pp

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