- Email: [email protected]
Chromatic spectrum is broken
Chromatic spectrum is broken Tao Jiang a, Dhruv Mubayi b, Zsolt Tuza c, Vitaly Voloshin d,1 and Douglas West e a Department b School
of Mathematics, University of Illinois, [email protected]
of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, [email protected]
c Computer d Institute
and Automation Institute, Hungarian Academy of Sciences, [email protected]
of Mathematics and Informatics, Moldovan Academy of Sciences, [email protected]
e Department
1
of Mathematics, University of Illinois, [email protected]
Introduction
A mixed hypergraph is a triple H = (X, C, D), where X is the vertex set and each of C, D is a family of subsets of X, the C-edges and D-edges, respectively. A proper k-coloring of a mixed hypergraph is a function from the vertex set to a set of k colors so that each C-edge has two vertices with a common color and each D-edge has two vertices with distinct colors. A mixed hypergraph is k-colorable if it has a proper coloring with at most k colors. A strict k-coloring is a proper k-coloring using all k colors. The minimum number of colors in a strict coloring of H is its lower chromatic number χ(H); the maximum number is its upper chromatic number χ(H). ¯ Introduced by the fourth author in [5], the theory of mixed hypergraphs is growing rapidly. It represents an area with many possible applications. They as models may be applied in list-free modeling of list-colorings of graphs [4], integer programming [4], investigating the coloring properties of block designs [3], in resource allocation, Data Base Management, parallel computing, scheduling of systems of power supplies and some other topics where the problems have combintorial nature [6]. A variant of a canonical Ramsey problem on edge-colorings studied in [2] and [1] can be phrased using coloring of mixed hypergraphs. 1
partially supported by COBASE (University of Illinois at Urbana-Champaign), University of Catania, DFG (TU-Dresden)
Preprint submitted to Elsevier Preprint
29 April 1999
We use n to denote |X| for the mixed hypergraph H = (X, C, D). For each k, let rk be the number of partitions of the vertex set into k nonempty parts (color classes) such that the coloring constraint is satisfied on each edge. The vector R(H) = (r1 , . . . , rn ) is the chromatic spectrum of H. The set of values k for which H has a strict k-coloring is the feasible set of H, written S(H); this is the set of indices i such that ri > 0. A mixed hypergraph H has lower chromatic number 1 if and only if H has no D-edges. In this case, color classes in a proper coloring can be combined to form a proper coloring using fewer colors, and thus S(H) = {1, . . . , χ(H)}. ¯ Similarly, χ(H) ¯ = n if and only if H has no C-edges. In this case, color classes in a proper coloring can be partitioned to form a proper coloring using more colors, and thus S(H) = {χ(H), . . . , n}. None of these procedures, however, is possible in general mixed hypergraphs. For all mixed hypergraphs whose spectra have been determined previously, the feasible set is the full interval from χ(H) to χ(H). ¯ The fourth author asked whether this is true for all mixed hypergraphs. We answer this in the negative and present optimal constructions of mixed hypergraphs having gaps in their chromatic spectra. The first examples of mixed hypergraphs with a gap in the chromatic spectrum was constructed by the second author. We say that a mixed hypergraph has a gap at k if its feasible set contains elements larger and smaller than k but omits k. In Section 2, we construct for 2 ≤ s ≤ t − 2 a mixed hypergraph Hs,t with feasible set {s, t}. Furthermore, we prove that Hs,t has the fewest vertices among all s-colorable mixed hypergraphs that have a gap at t − 1; this minimum number of vertices is 2t − s. This raises the question of which sets of positive integers are feasible sets of mixed hypergraphs. We solve this for finite sets in Section 3.
2
The Smallest Mixed Hypergraphs with Gaps
We begin with an explicit general construction for a mixed hypergraph with 2t − 2 vertices and feasible set {2, t}. Let Kn denote the mixed hypergraph with n vertices in which C = ∅ and D is the set of all pairs of vertices. Trivially, S(Kn ) = {n}. Our Construction 1 begins with Kt and expands t − 2 of the vertices into pairs, leaving two special vertices unexpanded. Each D-edge expands into a D-edge of size 3 or 4, depending on whether it contains one or two of the pairs. We add as C-edges all triples contained in each 4-element set of vertices formed by every two pairs. Lemma 1 The feasible set of the hypergraph H2,t in Construction 1 is {2, t}. In order to extend this construction to lower chromatic number s, we use a simple lemma about combining feasible sets. The join of two mixed hypergraphs 2
(X1 , C1 , D1 ) and (X2 , C2 , D2 ) with disjoint vertex sets is the mixed hypergraph (X, C, D) defined by X = X1 ∪ X2 , C = C1 ∪ C2 , and D = D1 ∪ D2 ∪ R, where R is the set of pairs consisting of one vertex from X1 and one from X2 . Lemma 2 If H1 and H2 are mixed hypergraphs, then the feasible set of the join of H1 and H2 is {i + j: i ∈ S(H1 ), j ∈ S(H2 )}. Theorem 3 If H is an s-colorable mixed hypergraph with a gap at t − 1, then n ≥ 2t − s, and this lower bound is tight for every s and t. Corollary 1 The minimum number of vertices in a mixed hypergraph with a gap in its feasible set is 6, achieved by H2,4 . The mixed hypergraph H2,4 is pictured on the Figure 1 below. The D-edges and coinciding C- and D-edges are drawn as classic edges, C-edges are drawn by dashed boxes. Coinciding C- and D-edges are called Bi-edges and crossed by the arrow. One can see that R(H2,4 ) = (0, 4, 0, 1, 0, 0).
s
s
Bi-edges ✯
s s s s
Figure 1. The smallest mixed hypergraph with gap H2,4 .
3
The Family of Feasible Sets
We characterize all finite sets that are feasible sets of mixed hypergraphs. The n-vertex mixed hypergraph (X, ∅, ∅) has feasible set {1, . . . , n}. This gives us all intervals containing 1, and we have already observed that these are the only possible feasible sets containing 1. 3
We construct mixed hypergraphs for all other feasible sets using these trivial mixed hypergraphs, the join operation of Lemma 2, and one additional operation. This operation is similar to the construction of H2,t from Kt . In that construction, we were able to avoid expanding two of the vertices, which we did in order to obtain the smallest number of vertices. Here our constructions will be huge, so we prefer the simplicity gained by expanding all vertices into pairs. Construction 2 Let H = (X, C, D) be a mixed hypergraph. We construct a mixed hypergraph H = (X , C , D ) with X = v∈X {v − , v +}. For each D ∈ D, we add D = v∈D {v − , v +} to D . For each C ∈ C, we add C = {v − : v ∈ C} to C . Finally, for each pair u, v ∈ X, we add the triples {v − , v +, u−}, {v −, v + , u+}, {u−, u+, v − } and {u−, u+, v + } to C . Lemma 4 Let H be a mixed hypergraph with feasible set S. If χ(H) ≥ 2, then the mixed hypergraph H obtained from H via Construction 2 has feasible set S ∪ {2}. Theorem 5 A finite set of positive integers is the feasible set for some mixed hypergraph if and only if it omits the number 1 or is an interval containing 1. References [1] T. Jiang and D. Mubayi, New upper bounds for a canonical Ramsey problem, Combinatorica, (to appear). [2] H. Lefmann, V. R¨ odl, and R. Thomas, Monochromatic vs multicolored paths, Graphs Combin., 8 (1992), 323–332. [3] L. Milazzo and Zs. Tuza, Upper chromatic number of Steiner Triple and Quadruple Systems, Discrete Math., 174 (1997) 247–259. [4] Zs. Tuza and V. Voloshin, Uncolorable mixed hypergraphs, Discrete Applied Math., (to appear). [5] V. Voloshin, On the upper chromatic number of a hypergraph, Australasian J. Comb., 11 (1995), 25–45. [6] V.Voloshin, Mixed hypergraphs as models for real problems, in preparation.
4
of Mathematics, University of Illinois, [email protected]
of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, [email protected]
c Computer d Institute
and Automation Institute, Hungarian Academy of Sciences, [email protected]
of Mathematics and Informatics, Moldovan Academy of Sciences, [email protected]
e Department
1
of Mathematics, University of Illinois, [email protected]
Introduction
A mixed hypergraph is a triple H = (X, C, D), where X is the vertex set and each of C, D is a family of subsets of X, the C-edges and D-edges, respectively. A proper k-coloring of a mixed hypergraph is a function from the vertex set to a set of k colors so that each C-edge has two vertices with a common color and each D-edge has two vertices with distinct colors. A mixed hypergraph is k-colorable if it has a proper coloring with at most k colors. A strict k-coloring is a proper k-coloring using all k colors. The minimum number of colors in a strict coloring of H is its lower chromatic number χ(H); the maximum number is its upper chromatic number χ(H). ¯ Introduced by the fourth author in [5], the theory of mixed hypergraphs is growing rapidly. It represents an area with many possible applications. They as models may be applied in list-free modeling of list-colorings of graphs [4], integer programming [4], investigating the coloring properties of block designs [3], in resource allocation, Data Base Management, parallel computing, scheduling of systems of power supplies and some other topics where the problems have combintorial nature [6]. A variant of a canonical Ramsey problem on edge-colorings studied in [2] and [1] can be phrased using coloring of mixed hypergraphs. 1
partially supported by COBASE (University of Illinois at Urbana-Champaign), University of Catania, DFG (TU-Dresden)
Preprint submitted to Elsevier Preprint
29 April 1999
We use n to denote |X| for the mixed hypergraph H = (X, C, D). For each k, let rk be the number of partitions of the vertex set into k nonempty parts (color classes) such that the coloring constraint is satisfied on each edge. The vector R(H) = (r1 , . . . , rn ) is the chromatic spectrum of H. The set of values k for which H has a strict k-coloring is the feasible set of H, written S(H); this is the set of indices i such that ri > 0. A mixed hypergraph H has lower chromatic number 1 if and only if H has no D-edges. In this case, color classes in a proper coloring can be combined to form a proper coloring using fewer colors, and thus S(H) = {1, . . . , χ(H)}. ¯ Similarly, χ(H) ¯ = n if and only if H has no C-edges. In this case, color classes in a proper coloring can be partitioned to form a proper coloring using more colors, and thus S(H) = {χ(H), . . . , n}. None of these procedures, however, is possible in general mixed hypergraphs. For all mixed hypergraphs whose spectra have been determined previously, the feasible set is the full interval from χ(H) to χ(H). ¯ The fourth author asked whether this is true for all mixed hypergraphs. We answer this in the negative and present optimal constructions of mixed hypergraphs having gaps in their chromatic spectra. The first examples of mixed hypergraphs with a gap in the chromatic spectrum was constructed by the second author. We say that a mixed hypergraph has a gap at k if its feasible set contains elements larger and smaller than k but omits k. In Section 2, we construct for 2 ≤ s ≤ t − 2 a mixed hypergraph Hs,t with feasible set {s, t}. Furthermore, we prove that Hs,t has the fewest vertices among all s-colorable mixed hypergraphs that have a gap at t − 1; this minimum number of vertices is 2t − s. This raises the question of which sets of positive integers are feasible sets of mixed hypergraphs. We solve this for finite sets in Section 3.
2
The Smallest Mixed Hypergraphs with Gaps
We begin with an explicit general construction for a mixed hypergraph with 2t − 2 vertices and feasible set {2, t}. Let Kn denote the mixed hypergraph with n vertices in which C = ∅ and D is the set of all pairs of vertices. Trivially, S(Kn ) = {n}. Our Construction 1 begins with Kt and expands t − 2 of the vertices into pairs, leaving two special vertices unexpanded. Each D-edge expands into a D-edge of size 3 or 4, depending on whether it contains one or two of the pairs. We add as C-edges all triples contained in each 4-element set of vertices formed by every two pairs. Lemma 1 The feasible set of the hypergraph H2,t in Construction 1 is {2, t}. In order to extend this construction to lower chromatic number s, we use a simple lemma about combining feasible sets. The join of two mixed hypergraphs 2
(X1 , C1 , D1 ) and (X2 , C2 , D2 ) with disjoint vertex sets is the mixed hypergraph (X, C, D) defined by X = X1 ∪ X2 , C = C1 ∪ C2 , and D = D1 ∪ D2 ∪ R, where R is the set of pairs consisting of one vertex from X1 and one from X2 . Lemma 2 If H1 and H2 are mixed hypergraphs, then the feasible set of the join of H1 and H2 is {i + j: i ∈ S(H1 ), j ∈ S(H2 )}. Theorem 3 If H is an s-colorable mixed hypergraph with a gap at t − 1, then n ≥ 2t − s, and this lower bound is tight for every s and t. Corollary 1 The minimum number of vertices in a mixed hypergraph with a gap in its feasible set is 6, achieved by H2,4 . The mixed hypergraph H2,4 is pictured on the Figure 1 below. The D-edges and coinciding C- and D-edges are drawn as classic edges, C-edges are drawn by dashed boxes. Coinciding C- and D-edges are called Bi-edges and crossed by the arrow. One can see that R(H2,4 ) = (0, 4, 0, 1, 0, 0).
s
s
Bi-edges ✯
s s s s
Figure 1. The smallest mixed hypergraph with gap H2,4 .
3
The Family of Feasible Sets
We characterize all finite sets that are feasible sets of mixed hypergraphs. The n-vertex mixed hypergraph (X, ∅, ∅) has feasible set {1, . . . , n}. This gives us all intervals containing 1, and we have already observed that these are the only possible feasible sets containing 1. 3
We construct mixed hypergraphs for all other feasible sets using these trivial mixed hypergraphs, the join operation of Lemma 2, and one additional operation. This operation is similar to the construction of H2,t from Kt . In that construction, we were able to avoid expanding two of the vertices, which we did in order to obtain the smallest number of vertices. Here our constructions will be huge, so we prefer the simplicity gained by expanding all vertices into pairs. Construction 2 Let H = (X, C, D) be a mixed hypergraph. We construct a mixed hypergraph H = (X , C , D ) with X = v∈X {v − , v +}. For each D ∈ D, we add D = v∈D {v − , v +} to D . For each C ∈ C, we add C = {v − : v ∈ C} to C . Finally, for each pair u, v ∈ X, we add the triples {v − , v +, u−}, {v −, v + , u+}, {u−, u+, v − } and {u−, u+, v + } to C . Lemma 4 Let H be a mixed hypergraph with feasible set S. If χ(H) ≥ 2, then the mixed hypergraph H obtained from H via Construction 2 has feasible set S ∪ {2}. Theorem 5 A finite set of positive integers is the feasible set for some mixed hypergraph if and only if it omits the number 1 or is an interval containing 1. References [1] T. Jiang and D. Mubayi, New upper bounds for a canonical Ramsey problem, Combinatorica, (to appear). [2] H. Lefmann, V. R¨ odl, and R. Thomas, Monochromatic vs multicolored paths, Graphs Combin., 8 (1992), 323–332. [3] L. Milazzo and Zs. Tuza, Upper chromatic number of Steiner Triple and Quadruple Systems, Discrete Math., 174 (1997) 247–259. [4] Zs. Tuza and V. Voloshin, Uncolorable mixed hypergraphs, Discrete Applied Math., (to appear). [5] V. Voloshin, On the upper chromatic number of a hypergraph, Australasian J. Comb., 11 (1995), 25–45. [6] V.Voloshin, Mixed hypergraphs as models for real problems, in preparation.
4