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On the representatives k-fold coloring polytope
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Electronic Notes in Discrete Mathematics 44 (2013) 239–244 www.elsevier.com/locate/endm
On the representatives k-fold coloring polytope Manoel Campˆelo a Phablo F. S. Moura b Marcio C. Santos c a
Dep. Estat´ıstica e Matem´ atica Aplicada, Universidade Federal do Cear´ a, Brazil b
Instituto de Matem´ atica e Estat´ıstica, Universidade de S˜ ao Paulo, Brazil c
Departamento de Computa¸c˜ ao, Universidade Federal do Cear´ a, Brazil
Abstract A k-fold x-coloring of a graph G is an assignment of (at least) k distinct colors from the set {1, 2, . . . , x} to each vertex such that any two adjacent vertices are assigned disjoint sets of colors. The k-th chromatic number of G, denoted by χk (G), is the smallest x such that G admits a k-fold x-coloring. We present an ILP formulation to determine χk (G) and study the facial structure of the corresponding polytope Pk (G). We show facets that Pk+1 (G) inherits from Pk (G). We also relate Pk (G) to P1 (G◦Kk ), where G◦Kk is the lexicographic product of G by a clique with k vertices. In both cases, we can obtain facet-defining inequalities from many of those known for the 1-fold coloring polytope. In addition, we present a class of facet-defining inequalities based on strongly χk -critical webs, which extend and generalize known corresponding results for 1-fold coloring. We introduce this criticality concept and characerize the webs having such a property. Keywords: (k-fold) graph coloring, facet, web graph, critical graph, lexicographic product.
1
Research supported by CNPq (Proc. 307627/2010-1, 480608/2011-3, 132998/2011-4), Capes, and Projects FUNCAP/PRONEM and STIC-AmSud. 2 E-mail address: [email protected], [email protected], [email protected]. 1571-0653/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.endm.2013.10.037
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M. Campêlo et al. / Electronic Notes in Discrete Mathematics 44 (2013) 239–244
The k-fold coloring problem
A k-fold x-coloring of a graph G is an assignment of (at least) k distinct colors from the set [x] := {1, 2, . . . , x} to each vertex such that any two adjacent vertices are assigned disjoint sets of colors. The k-th chromatic number of G, denoted χk (G), is the smallest value x such that G admits a k-fold xcoloring [6]. Obviously, χ1 (G) = χ(G) is the conventional chromatic number. Each color class (vertices with the same color) defines a stable set of the graph (subset of pairwise nonadjacent vertices). Therefore, a coloring is a covering of the vertices by stable sets. A family of stable sets can be described ¯ = (V, E) ¯ be by the idea of class representatives [1,2]. Let G = (V, E) and G the complement of G. Given an ordering on V , the representative of a stable set S is its minimum vertex r (which represents each vertex in S), and S can be uniquely described by a binary vector xS ∈ Bn+m¯ , indexed by the n vertices ¯ having 1 in the entries corresponding to r ∈ S and rv ∈ E, ¯ and m ¯ edges of G, for each v ∈ S \ r. This yields the idea to model the k-fold coloring problem. ¯ − (v) = {u ≺ v | uv ∈ E} ¯ More precisely, let ≺ be an ordering on V . Let N + ¯ (v) = {v ≺ u | uv ∈ E} ¯ be the negative and postive anti-neighborhood and N ¯ ¯ − (v) ∪ {v}. For each i ∈ [k], v ∈ V and of v, respectively. Denote N − [v] = N − ¯ [v], define a binary variable xi to be set 1 iff the i-th color class u ∈ N uv repesented by u is nonempty and includes v. Notice that we are considering that each vertex v represents k (possibly empty) color classes, and xivv = 1 marks that the i-th class v represents is nonempty. The representatives formulation for 1-fold coloring [1] can be easily generalized for k-fold coloring: Rk (G) min s.t.
k
xivv
i=1 v∈V k
(1) xiuv ≥ k ∀v ∈ V
¯ − [v] i=1 u∈N i xvu + xivw ≤ xivw ≤ xivv xiuv ∈ {0, 1}
xivv
(2)
¯ + (v), uw ∈ E, ∀i ∈ [k], (3) ∀v ∈ V, ∀u, w ∈ N ¯ + (v), ∀i ∈ [k], ∀v ∈ V, ∀w ∈ N (4) − ¯ [v], ∀i ∈ [k]. ∀v ∈ V, ∀u ∈ N (5)
Actually, a k-fold coloring of G is equivalent to an 1-fold coloring of the graph G◦Kk , the lexicographic product of G by a clique with k vertices. Recall that G ◦ Kk is obtained by replacing each vertex of G by the clique Kk and making two vertices in G ◦ Kk adjacent whenever the corresponding vertices in G are adjacent. Instead of using the formulation R1 (G ◦ Kk ), we propose the
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more compact formulation Rk (G). However, we do not disregard the many known results for the (1-fold) coloring polytope. First, we define Pk (G), the polytope associated with Rk (G). In the Section 3, we show how to lift facets from Pk (G) to Pk+1 (G). In the Section 4, we show how to project facets from P1 (G ◦ Kk ) to Pk (G). In both cases, we can recover most of the facet-defining inequalities of the coloring polytope from [1]. Moreover, we derive a class of facet-inducing inequalities based on webs, which are structures that play an important role in the description of stable set and coloring polytopes [4,8]. In this case, we extend and generalize the results in [1,5].
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The k-fold coloring polytope
¯ − (v) = ∅} be the set of sources and n := n − |M |. By (2), we Let M = {v | N have that xivv = 1 ∀v ∈ M and ∀i ∈ [k]. Since these variables can be removed from Rk (G), we define its corresponding polytope as ¯ ¯ | (1, x) ∈ Bk(n+m) satisfies (2) − (5)}, Pk (G) = conv{x ∈ Bk(n +m) where vector 1 comprises the entries indexed by i ∈ [k] and v ∈ M . For a fixed i ∈ [k], the convex hull of the points satisfying (3)-(5) and u∈N¯ − [v] xiuv ≥ 1 is equal to P1 (G). So, Pk (G) ⊇ P1 (G) × · · · × P1 (G) (k times). If v is universal ¯ (v) := N ¯ − (v) ∪ N ¯ + (v) = ∅), then χk (G) = χk (G − v) + k. If v is dominated (N ¯ ¯ (N [v] ⊇ N [u] for some u ∈ V ), then χk (G) = χk (G − v). Henceforth, we assume that G has no universal nor dominated vertex. Thus, P1 (G) is fulldimensional [1]. Following the results in [1], we can obtain: Proposition 2.1 Let k ∈ N. Pk (G) is full-dimensional and, for all i ∈ [k], ¯ + (u) = ∅; has a facet induced by: xiuu ≤ 1, ∀u ∈ V \ M ; xiuu ≥ 0, ∀u ∈ V , N i − ¯ (v)| ≥ 2 if k = 1, and ∀u ∈ N ¯ − (v). In xuv ≥ 0, ∀v ∈ V \ M , with |N addition, inequality (2) defines a facet of Pk (G). Given x ∈ Pk (G) and i ∈ [k], let xi denote the subvector of x defined by ¯ − [v]. Since interchanging the values of the entries xiuv , for v ∈ V \ M , u ∈ N j x and x yields another point of Pk (G), we get Proposition 2.2 Let k ∈ N and j, ∈ [k]. If i∈[k] λi xi ≤ λ0 defines a facet of Pk (G), then i∈[k]\{j,} λi xi + λj x + λ xj ≤ λ0 defines a facet of Pk (G).
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Facets from Pk (G) to Pk+1 (G)
Some inequalities defining facets of Pk (G) can be trivially lifted to generate facets of Pk+1 . Sometimes, we can carry these inequalities since k = 1.
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¯ Theorem 3.1 Let k ∈ N, (λ, λ0 ) ∈ Rk(n +m) × R be a facet-defining inequality n +m ¯ of Pk (G), and 0 ∈ R . If (λ, 0 , λ0 ) is a valid inequality of Pk+1 (G), then (λ, 0 , λ0 ) defines a facet of Pk+1 (G).
¯ } be a set of affinely independent Proof. [Sketch] Let Y = {y 1 , . . . , y k(n +m) vectors in the facet of Pk (G) defined by (λ, λ0 ). Let V \ M = {v1 , v2 , . . . , vn } and E¯ = {a1 , a2 , . . . , am¯ }. For every i ∈ [m], ¯ denote by ei ∈ Bm¯ the unit vector with 1 in the entry associated with ai . For every i ∈ [n ], denote by e¯i ∈ Bn the vector with 0 only in the entry associated with vi , and let ar(i) ∈ E¯ be an arbitrary edge with endpoints vi and v , where v ≺ vi . We define the ¯ sets of vectors Z1 := {(y i , 1 , 0 ) ∈ Bk(n +m) × Bn × Bm¯ | i ∈ [k(n + m)]}, ¯ i i k(n +m) ¯ n m ¯ m+i ¯ i r(i) Z2 := {(y , 1 , e ) ∈ B × B × B | i ∈ [m]} ¯ and Z3 := {(y , e¯ , e ) ∈ ¯ Bk(n +m) × Bn × Bm¯ | i ∈ [n ]}. We can show Z1 ∪ Z2 ∪ Z3 is a set of affinely independent points in the face of Pk+1 (G) defined by (λ, 0 , λ0 ). 2
¯ + (u). For v ∈ H, let αv be the maxCorollary 3.2 Let u ∈ V and H ⊆ N imum size of a stable set of G[H] containing v, where G[H] is the subgraph of by the vertices in H. For k ∈ N and i ∈ [k], let Θki (x) := G induced 1 i i i i i v∈H αv xuv − yu , where yu = 1, if u ∈ M , and yu = xuu , otherwise. The inequality Θki (x) ≤ 0 is valid for Pk (G). It induces a facet of Pk (G) under the same conditions Θ11 (x) ≤ 0 induces a facet of P1 (G) (see [1, Theorem 2]).
4
Facets from P1 (G ◦ Kk ) to Pk (G)
For each v ∈ V = V (G) and i ∈ [k], let vi ∈ V ◦ = V (G ◦ Kk ) denote the i-th copy of v in G ◦ Kk . Given the total order ≺ on V , let us define a total order ˙ j , for every v ∈ V and i < j, and ui ≺v ˙ j , for every ˙ on V ◦ such that vi ≺v ≺ − − ¯ ¯ u, v ∈ V with u ≺ v. This way, if u ∈ N (v), then ui ∈ N (vj ) ∀i, j ∈ [k]. If v is a source in G, then v1 , v2 , . . . , vk are sources in G ◦ Kk . ¯ We observe that Pk (G) ⊆ Rp whereas P1 (G◦Kk ) ⊆ Rq , where p = k(n +m) and q = k(n + k m). ¯ As k increases, the difference q − p increases. This is a reason to explore formulation Rk (G) instead of simply using R1 (G ◦ Kk ). However, we profit from the known results for P1 (G ◦ Kk ) to describe Pk (G). Theorem 4.1 Let (λ, λ0 ) ∈ Rq × R induce a facet of P1 (G ◦ Kk ). If, for ¯ − (v) and i ∈ [k], there is λu v such that λu v = λu v every v ∈ V \ M , u ∈ N i i i j ˆ λ0 ) ∈ Rp × R, where ∀j ∈ [k], then a facet-inducing inequality of Pk (G) is (λ, ˆ i = λv v , λ ˆ i = λu v , for all v ∈ V \ M , u ∈ N ¯ − (v) and i ∈ [k]. λ vv uv i i i (G), where y = f (x) is Proof. [Sketch] Define a function f : P1 (G ◦ Kk ) → Pk i i such that yuu = xui ui ∀u ∈ V \ M , ∀i ∈ [k], and yuv = kj=1 xui vj ∀v ∈ V \ M ,
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¯ − (v), ∀i ∈ [k]. Is not hard to see that f is an onto function and ∀u ∈ N f (x) = Ax for a p × q 0-1 matrix with rank(A) = p. Let X be a set of q linearly independent vectors in the facet defined by (λ, λ0 ). Since rank(A) = p, {Ax | x ∈ X} contains p linearly independent vectors in Pk (G). Moreover, as ˆ T A = λT , these vectors lie on the facet defined by (λ, ˆ λ ˆ 0 ). λ 2
5
Facets induced by webs
For integers p ≥ 1 and n ≥ 2p, Trotter [7] defines the web Wpn as the graph with vertex set {v0 , v1 , . . . , vn−1 } and edge set {vi vj | p ≤ |i − j| ≤ n − p}. Webs form a subclass of circulant graphs that play an important role in the context of stable sets and vertex coloring problems. They include odd holes (W2+1 ), odd anti-holes (W22+1 ) and cliques (W1n ). Facet-defining inequalities of P1 (G) based on odd holes and odd anti-holes are presented in [1]. More recently, other webs are considered to obtain facets of an 1-fold coloring polytope associated with a formulation similar to R1 (G) [5]. Here, we generalize these results. For every k ∈ N, a graph G is χk critical if χk (G − v) < χk (G), for each vertex v. χk -critical webs are characterized in [3]. Here, we introduce a stronger criticality condition that plays a role in facet-defining structures. We will say that G is strongly χk -critical if, for every v ∈ V , G admits an optimal k-fold coloring where v forms a color class. Proposition 5.1 For every k ∈ N, Wpn is strongly χk -critical if, and only if, nk−1 ∈ N. p Proof. [Sketch] In any optimal k-fold coloring of Wpn , we have χk (Wpn ) = nk color classes, each of them with at most p vertices [3]. Then, if one of p them is a singleton, the other ones must have exactly p vertices. Therefore, − 1)p = nk − 1, implying that nk−1 ∈ N. Conversely, assume that ( nk p p
x = nk−1 ∈ N. By traversing the sequence v0 , v1 , . . . vn−1 in a circular way, p define x subsets of p vertices each. Each subset forms a color class [3]. Morever, they cover n − 1 vertices k times and 1 vertex k − 1 times. Using this last nk +1 = , vertex to form an additional color class, we get a coloring with nk−1 p p which is optimal [3]. Therefore, Wpn is strongly χk -critical. 2 Theorem 5.2 Let k ∈ N, H ⊆ V and MH be the set of sources in ≺ restricted to H. If G[H], i.e. the subgraph of G induced by H, is a strongly χk -critical web that is not a clique, then a facet of Pk (G) is induced by the inequality
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k
¯ − (v)\H)∪{v} i=1 v∈H\MH u∈(N
xiuv ≥ χk (G[H]) − k|MH |.
(6)
Proof. [Sketch] It is not hard to see that (6) is valid for Pk (G). Indeed, since MH induces a clique in G, we need at least χk (G[H]) − k|MH | colors different from those appearing in MH to color the vertices in H \ MH . The proof that it induces a facet follows a strategy similar to that one used in [1, Theorem ˆT x ≥ λ ˆ 0 correspond to (6), Fˆ = {x ∈ Pk (G) | λ ˆT x = λ ˆ 0 }, and 3]. Let λ T ˆ F = {x ∈ Pk (G) | λ x = λ0 } be a face of Pk (G) containing F . We prove ˆ λ ˆ 0 ), for some α > 0. We first determine the null entries of that (λ, λ0 ) = α(λ, ¯ − (v) \ H and λ. Then, we prove that λjuv = λivv =: λv ∀v ∈ H \ MH , ∀u ∈ N ∀i, j ∈ [k]. Using induction on ≺, we show that λv = λw , for all v, w ∈ H \MH ¯ − (w). If G[H ¯ \ MH ] is connected, the remaining equalities follow with v ∈ N ¯ is a hole, and a specific analysis is carried out by connectivity. Otherwise, G to get the same equalities. 2
References [1] Campˆelo, M., V. Campos and R. Corrˆea, On the Asymmetric Representatives Formulation for the Vertex Coloring Problem, Discrete Applied Mathematics 156 (2008), pp. 1097–1111. [2] Campˆelo, M., R. Corrˆea and Y. Frota, Cliques, holes and the vertex coloring polytope, Information Processing Letters 89 (2004), pp. 159–164. [3] Campˆelo, M., R. Corrˆea, P. Moura and M. C. Santos, On optimal k-fold colorings of webs and antiwebs, Discrete Applied Mathematics 161 (2013), pp. 60–70. [4] Cheng, E. and S. de Vries, On the facet-inducing antiweb-wheel inequalities for stable set polytopes, SIAM J. on Discrete Mathematics 15 (2002), pp. 470–487. [5] Palubeckis, G., Facet-inducing web and antiweb inequalities for the graph coloring polytope, Discrete Applied Mathematics 158 (2010), pp. 2075–2080. [6] Stahl, S., n-Tuple colorings and associated graphs, Journal of Combinatorial Theory 20 (1976), pp. 185–203. [7] Trotter, L. E., A Class of facet producing graphs for vertex packing polyhedra, Discrete Mathematics 12 (1975), pp. 373–388. [8] Wagler, A. K., Antiwebs are rank-perfect, 4OR: A Quarterly Journal of Operations Research 2 (2004), pp. 149–152.
Electronic Notes in Discrete Mathematics 44 (2013) 239–244 www.elsevier.com/locate/endm
On the representatives k-fold coloring polytope Manoel Campˆelo a Phablo F. S. Moura b Marcio C. Santos c a
Dep. Estat´ıstica e Matem´ atica Aplicada, Universidade Federal do Cear´ a, Brazil b
Instituto de Matem´ atica e Estat´ıstica, Universidade de S˜ ao Paulo, Brazil c
Departamento de Computa¸c˜ ao, Universidade Federal do Cear´ a, Brazil
Abstract A k-fold x-coloring of a graph G is an assignment of (at least) k distinct colors from the set {1, 2, . . . , x} to each vertex such that any two adjacent vertices are assigned disjoint sets of colors. The k-th chromatic number of G, denoted by χk (G), is the smallest x such that G admits a k-fold x-coloring. We present an ILP formulation to determine χk (G) and study the facial structure of the corresponding polytope Pk (G). We show facets that Pk+1 (G) inherits from Pk (G). We also relate Pk (G) to P1 (G◦Kk ), where G◦Kk is the lexicographic product of G by a clique with k vertices. In both cases, we can obtain facet-defining inequalities from many of those known for the 1-fold coloring polytope. In addition, we present a class of facet-defining inequalities based on strongly χk -critical webs, which extend and generalize known corresponding results for 1-fold coloring. We introduce this criticality concept and characerize the webs having such a property. Keywords: (k-fold) graph coloring, facet, web graph, critical graph, lexicographic product.
1
Research supported by CNPq (Proc. 307627/2010-1, 480608/2011-3, 132998/2011-4), Capes, and Projects FUNCAP/PRONEM and STIC-AmSud. 2 E-mail address: [email protected], [email protected], [email protected]. 1571-0653/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.endm.2013.10.037
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M. Campêlo et al. / Electronic Notes in Discrete Mathematics 44 (2013) 239–244
The k-fold coloring problem
A k-fold x-coloring of a graph G is an assignment of (at least) k distinct colors from the set [x] := {1, 2, . . . , x} to each vertex such that any two adjacent vertices are assigned disjoint sets of colors. The k-th chromatic number of G, denoted χk (G), is the smallest value x such that G admits a k-fold xcoloring [6]. Obviously, χ1 (G) = χ(G) is the conventional chromatic number. Each color class (vertices with the same color) defines a stable set of the graph (subset of pairwise nonadjacent vertices). Therefore, a coloring is a covering of the vertices by stable sets. A family of stable sets can be described ¯ = (V, E) ¯ be by the idea of class representatives [1,2]. Let G = (V, E) and G the complement of G. Given an ordering on V , the representative of a stable set S is its minimum vertex r (which represents each vertex in S), and S can be uniquely described by a binary vector xS ∈ Bn+m¯ , indexed by the n vertices ¯ having 1 in the entries corresponding to r ∈ S and rv ∈ E, ¯ and m ¯ edges of G, for each v ∈ S \ r. This yields the idea to model the k-fold coloring problem. ¯ − (v) = {u ≺ v | uv ∈ E} ¯ More precisely, let ≺ be an ordering on V . Let N + ¯ (v) = {v ≺ u | uv ∈ E} ¯ be the negative and postive anti-neighborhood and N ¯ ¯ − (v) ∪ {v}. For each i ∈ [k], v ∈ V and of v, respectively. Denote N − [v] = N − ¯ [v], define a binary variable xi to be set 1 iff the i-th color class u ∈ N uv repesented by u is nonempty and includes v. Notice that we are considering that each vertex v represents k (possibly empty) color classes, and xivv = 1 marks that the i-th class v represents is nonempty. The representatives formulation for 1-fold coloring [1] can be easily generalized for k-fold coloring: Rk (G) min s.t.
k
xivv
i=1 v∈V k
(1) xiuv ≥ k ∀v ∈ V
¯ − [v] i=1 u∈N i xvu + xivw ≤ xivw ≤ xivv xiuv ∈ {0, 1}
xivv
(2)
¯ + (v), uw ∈ E, ∀i ∈ [k], (3) ∀v ∈ V, ∀u, w ∈ N ¯ + (v), ∀i ∈ [k], ∀v ∈ V, ∀w ∈ N (4) − ¯ [v], ∀i ∈ [k]. ∀v ∈ V, ∀u ∈ N (5)
Actually, a k-fold coloring of G is equivalent to an 1-fold coloring of the graph G◦Kk , the lexicographic product of G by a clique with k vertices. Recall that G ◦ Kk is obtained by replacing each vertex of G by the clique Kk and making two vertices in G ◦ Kk adjacent whenever the corresponding vertices in G are adjacent. Instead of using the formulation R1 (G ◦ Kk ), we propose the
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more compact formulation Rk (G). However, we do not disregard the many known results for the (1-fold) coloring polytope. First, we define Pk (G), the polytope associated with Rk (G). In the Section 3, we show how to lift facets from Pk (G) to Pk+1 (G). In the Section 4, we show how to project facets from P1 (G ◦ Kk ) to Pk (G). In both cases, we can recover most of the facet-defining inequalities of the coloring polytope from [1]. Moreover, we derive a class of facet-inducing inequalities based on webs, which are structures that play an important role in the description of stable set and coloring polytopes [4,8]. In this case, we extend and generalize the results in [1,5].
2
The k-fold coloring polytope
¯ − (v) = ∅} be the set of sources and n := n − |M |. By (2), we Let M = {v | N have that xivv = 1 ∀v ∈ M and ∀i ∈ [k]. Since these variables can be removed from Rk (G), we define its corresponding polytope as ¯ ¯ | (1, x) ∈ Bk(n+m) satisfies (2) − (5)}, Pk (G) = conv{x ∈ Bk(n +m) where vector 1 comprises the entries indexed by i ∈ [k] and v ∈ M . For a fixed i ∈ [k], the convex hull of the points satisfying (3)-(5) and u∈N¯ − [v] xiuv ≥ 1 is equal to P1 (G). So, Pk (G) ⊇ P1 (G) × · · · × P1 (G) (k times). If v is universal ¯ (v) := N ¯ − (v) ∪ N ¯ + (v) = ∅), then χk (G) = χk (G − v) + k. If v is dominated (N ¯ ¯ (N [v] ⊇ N [u] for some u ∈ V ), then χk (G) = χk (G − v). Henceforth, we assume that G has no universal nor dominated vertex. Thus, P1 (G) is fulldimensional [1]. Following the results in [1], we can obtain: Proposition 2.1 Let k ∈ N. Pk (G) is full-dimensional and, for all i ∈ [k], ¯ + (u) = ∅; has a facet induced by: xiuu ≤ 1, ∀u ∈ V \ M ; xiuu ≥ 0, ∀u ∈ V , N i − ¯ (v)| ≥ 2 if k = 1, and ∀u ∈ N ¯ − (v). In xuv ≥ 0, ∀v ∈ V \ M , with |N addition, inequality (2) defines a facet of Pk (G). Given x ∈ Pk (G) and i ∈ [k], let xi denote the subvector of x defined by ¯ − [v]. Since interchanging the values of the entries xiuv , for v ∈ V \ M , u ∈ N j x and x yields another point of Pk (G), we get Proposition 2.2 Let k ∈ N and j, ∈ [k]. If i∈[k] λi xi ≤ λ0 defines a facet of Pk (G), then i∈[k]\{j,} λi xi + λj x + λ xj ≤ λ0 defines a facet of Pk (G).
3
Facets from Pk (G) to Pk+1 (G)
Some inequalities defining facets of Pk (G) can be trivially lifted to generate facets of Pk+1 . Sometimes, we can carry these inequalities since k = 1.
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¯ Theorem 3.1 Let k ∈ N, (λ, λ0 ) ∈ Rk(n +m) × R be a facet-defining inequality n +m ¯ of Pk (G), and 0 ∈ R . If (λ, 0 , λ0 ) is a valid inequality of Pk+1 (G), then (λ, 0 , λ0 ) defines a facet of Pk+1 (G).
¯ } be a set of affinely independent Proof. [Sketch] Let Y = {y 1 , . . . , y k(n +m) vectors in the facet of Pk (G) defined by (λ, λ0 ). Let V \ M = {v1 , v2 , . . . , vn } and E¯ = {a1 , a2 , . . . , am¯ }. For every i ∈ [m], ¯ denote by ei ∈ Bm¯ the unit vector with 1 in the entry associated with ai . For every i ∈ [n ], denote by e¯i ∈ Bn the vector with 0 only in the entry associated with vi , and let ar(i) ∈ E¯ be an arbitrary edge with endpoints vi and v , where v ≺ vi . We define the ¯ sets of vectors Z1 := {(y i , 1 , 0 ) ∈ Bk(n +m) × Bn × Bm¯ | i ∈ [k(n + m)]}, ¯ i i k(n +m) ¯ n m ¯ m+i ¯ i r(i) Z2 := {(y , 1 , e ) ∈ B × B × B | i ∈ [m]} ¯ and Z3 := {(y , e¯ , e ) ∈ ¯ Bk(n +m) × Bn × Bm¯ | i ∈ [n ]}. We can show Z1 ∪ Z2 ∪ Z3 is a set of affinely independent points in the face of Pk+1 (G) defined by (λ, 0 , λ0 ). 2
¯ + (u). For v ∈ H, let αv be the maxCorollary 3.2 Let u ∈ V and H ⊆ N imum size of a stable set of G[H] containing v, where G[H] is the subgraph of by the vertices in H. For k ∈ N and i ∈ [k], let Θki (x) := G induced 1 i i i i i v∈H αv xuv − yu , where yu = 1, if u ∈ M , and yu = xuu , otherwise. The inequality Θki (x) ≤ 0 is valid for Pk (G). It induces a facet of Pk (G) under the same conditions Θ11 (x) ≤ 0 induces a facet of P1 (G) (see [1, Theorem 2]).
4
Facets from P1 (G ◦ Kk ) to Pk (G)
For each v ∈ V = V (G) and i ∈ [k], let vi ∈ V ◦ = V (G ◦ Kk ) denote the i-th copy of v in G ◦ Kk . Given the total order ≺ on V , let us define a total order ˙ j , for every v ∈ V and i < j, and ui ≺v ˙ j , for every ˙ on V ◦ such that vi ≺v ≺ − − ¯ ¯ u, v ∈ V with u ≺ v. This way, if u ∈ N (v), then ui ∈ N (vj ) ∀i, j ∈ [k]. If v is a source in G, then v1 , v2 , . . . , vk are sources in G ◦ Kk . ¯ We observe that Pk (G) ⊆ Rp whereas P1 (G◦Kk ) ⊆ Rq , where p = k(n +m) and q = k(n + k m). ¯ As k increases, the difference q − p increases. This is a reason to explore formulation Rk (G) instead of simply using R1 (G ◦ Kk ). However, we profit from the known results for P1 (G ◦ Kk ) to describe Pk (G). Theorem 4.1 Let (λ, λ0 ) ∈ Rq × R induce a facet of P1 (G ◦ Kk ). If, for ¯ − (v) and i ∈ [k], there is λu v such that λu v = λu v every v ∈ V \ M , u ∈ N i i i j ˆ λ0 ) ∈ Rp × R, where ∀j ∈ [k], then a facet-inducing inequality of Pk (G) is (λ, ˆ i = λv v , λ ˆ i = λu v , for all v ∈ V \ M , u ∈ N ¯ − (v) and i ∈ [k]. λ vv uv i i i (G), where y = f (x) is Proof. [Sketch] Define a function f : P1 (G ◦ Kk ) → Pk i i such that yuu = xui ui ∀u ∈ V \ M , ∀i ∈ [k], and yuv = kj=1 xui vj ∀v ∈ V \ M ,
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¯ − (v), ∀i ∈ [k]. Is not hard to see that f is an onto function and ∀u ∈ N f (x) = Ax for a p × q 0-1 matrix with rank(A) = p. Let X be a set of q linearly independent vectors in the facet defined by (λ, λ0 ). Since rank(A) = p, {Ax | x ∈ X} contains p linearly independent vectors in Pk (G). Moreover, as ˆ T A = λT , these vectors lie on the facet defined by (λ, ˆ λ ˆ 0 ). λ 2
5
Facets induced by webs
For integers p ≥ 1 and n ≥ 2p, Trotter [7] defines the web Wpn as the graph with vertex set {v0 , v1 , . . . , vn−1 } and edge set {vi vj | p ≤ |i − j| ≤ n − p}. Webs form a subclass of circulant graphs that play an important role in the context of stable sets and vertex coloring problems. They include odd holes (W2+1 ), odd anti-holes (W22+1 ) and cliques (W1n ). Facet-defining inequalities of P1 (G) based on odd holes and odd anti-holes are presented in [1]. More recently, other webs are considered to obtain facets of an 1-fold coloring polytope associated with a formulation similar to R1 (G) [5]. Here, we generalize these results. For every k ∈ N, a graph G is χk critical if χk (G − v) < χk (G), for each vertex v. χk -critical webs are characterized in [3]. Here, we introduce a stronger criticality condition that plays a role in facet-defining structures. We will say that G is strongly χk -critical if, for every v ∈ V , G admits an optimal k-fold coloring where v forms a color class. Proposition 5.1 For every k ∈ N, Wpn is strongly χk -critical if, and only if, nk−1 ∈ N. p Proof. [Sketch] In any optimal k-fold coloring of Wpn , we have χk (Wpn ) = nk color classes, each of them with at most p vertices [3]. Then, if one of p them is a singleton, the other ones must have exactly p vertices. Therefore, − 1)p = nk − 1, implying that nk−1 ∈ N. Conversely, assume that ( nk p p
x = nk−1 ∈ N. By traversing the sequence v0 , v1 , . . . vn−1 in a circular way, p define x subsets of p vertices each. Each subset forms a color class [3]. Morever, they cover n − 1 vertices k times and 1 vertex k − 1 times. Using this last nk +1 = , vertex to form an additional color class, we get a coloring with nk−1 p p which is optimal [3]. Therefore, Wpn is strongly χk -critical. 2 Theorem 5.2 Let k ∈ N, H ⊆ V and MH be the set of sources in ≺ restricted to H. If G[H], i.e. the subgraph of G induced by H, is a strongly χk -critical web that is not a clique, then a facet of Pk (G) is induced by the inequality
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k
¯ − (v)\H)∪{v} i=1 v∈H\MH u∈(N
xiuv ≥ χk (G[H]) − k|MH |.
(6)
Proof. [Sketch] It is not hard to see that (6) is valid for Pk (G). Indeed, since MH induces a clique in G, we need at least χk (G[H]) − k|MH | colors different from those appearing in MH to color the vertices in H \ MH . The proof that it induces a facet follows a strategy similar to that one used in [1, Theorem ˆT x ≥ λ ˆ 0 correspond to (6), Fˆ = {x ∈ Pk (G) | λ ˆT x = λ ˆ 0 }, and 3]. Let λ T ˆ F = {x ∈ Pk (G) | λ x = λ0 } be a face of Pk (G) containing F . We prove ˆ λ ˆ 0 ), for some α > 0. We first determine the null entries of that (λ, λ0 ) = α(λ, ¯ − (v) \ H and λ. Then, we prove that λjuv = λivv =: λv ∀v ∈ H \ MH , ∀u ∈ N ∀i, j ∈ [k]. Using induction on ≺, we show that λv = λw , for all v, w ∈ H \MH ¯ − (w). If G[H ¯ \ MH ] is connected, the remaining equalities follow with v ∈ N ¯ is a hole, and a specific analysis is carried out by connectivity. Otherwise, G to get the same equalities. 2
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