Local chromatic number and Sperner capacity

Introduction

Colouring the vertices of a graph so that no adjacent vertices receive identical colours gives rise to many interesting problems and invariants, of which the book [17] gives an excellent survey. The best known among all these invariants is the chromatic number, the minimum number of colours needed for such proper colourings. An interesting variant was E-mail addresses: korner@di.uniroma1.it (J. Körner), pilotto@di.uniroma1.it (C. Pilotto), simonyi@renyi.hu (G. Simonyi). 1 introduced by Erdős et al. [10] (cf. also [12]). They define the local chromatic number of a graph as follows.

Definition 1 (Erdős et al. [10]). The local chromatic number (G) of a graph G is the maximum number of different colours appearing in the closed neighbourhood of any vertex, minimized over all proper colourings of G. Formally,

where N is the set of natural numbers, G (v), the closed neighborhood of the vertex v ∈ V (G), is the set of those vertices of G that are either adjacent or equal to v and c : V (G) → N runs over those functions that are proper colourings of G.

It is clear that (G) is always bounded from above by the chromatic number, (G). It is much less obvious that (G) can be strictly less than (G). Yet this is true; in fact, as proved in [10], there exist graphs with (G) = 3 and (G) arbitrarily large.

Throughout this paper, we shall be interested in chromatic invariants as upper bounds for the Shannon capacity of undirected graphs and its natural generalization, Sperner capacity, for directed graphs. For the sake of unity in the treatment of undirected and directed graphs it is convenient and customary to treat Shannon capacity in terms that are complementary to Shannon's own, (cf. [24,20,14,18]). In this language Shannon capacity describes the asymptotic growth of the clique number in the co-normal powers of a graph. Shannon proved (although in different terms) that the Shannon capacity c(G) of a graph is bounded from above by its fractional chromatic number.

We show that (G) is bounded from below by the fractional chromatic number of G. This proves, among other things, that (G) is always an upper bound for the Shannon capacity c(G) of G, but it is not a very useful upper bound since it is always weaker than the fractional chromatic number itself. We make this seemingly useless remark only to stress that the situation is rather different in the case of directed graphs.

We introduce an analog of the local chromatic number for directed graphs and show that it is always an upper bound for the Sperner capacity of the digraph at hand. The proof is linear algebraic and generalizes an idea already used for bounding Sperner capacity in [6,1,11], cf. also [8]. To illustrate the usefulness of this bound we apply it to show, for example, that an oriented odd cycle with at least two vertices with outdegree and indegree 1 always has its Sperner capacity equal to that of the single-edge graph K 2 . We also discuss fractional versions that further strengthen our bounds.

Local chromatic number for directed graphs

The definition of the directed version of (G) is straightforward.

where N is the set of natural numbers, + G (v), the closed out-neighborhood of the vertex v ∈ V (G), is the set of those vertices w ∈ V (G) that are either equal to v or else are endpoints of directed edges (v, w) ∈ E(G), originated in v, and c : V (G) → N runs over those functions that are proper colourings of G.

Our main goal is to prove that d (G) is an upper bound for the Sperner capacity of digraph G.

Sperner capacity Definition For directed graphs G = (V , E)

and H = (W, L), the co-normal (or disjunctive or OR) product G • H is defined to be the following directed graph:

The nth co-normal (or disjunctive or OR) power G n of digraph G is defined as the n-fold co-normal product of G with itself, i.e., the vertex set of G n is V n = {x = (x 1 , . . . , x n ) :

(An edge (a, b) always means an oriented edge in this paper as opposed to undirected edges denoted by {a, b}.)

Notice that G n may contain edges in both directions between two vertices even if such a pair of edges is not present in G.

Definition 4.

A subgraph of a digraph is called a symmetric clique if its edge set contains all ordered pairs of vertices belonging to the subgraph. (In other words, it is a clique with all its edges present in both directions.) For a directed graph G we denote the order (number of vertices) of its largest symmetric clique by s (G). Definition 5 (Gargano et al. [14]). The (non-logarithmic) Sperner capacity of a digraph G is defined as

Remark. Denoting the number of vertices in a largest transitive clique of G by tr(G), it is easy to show that (G) = sup n n √ tr(G n ) holds, cf. [14,22] and the references therein. (By a transitive clique we mean a clique where the edges are oriented transitively, i.e., consistently with some linear order of the vertices. It is allowed that some edges be present also in the reverse direction.) Since tr(G n ) [tr(G)] n this remark implies that tr(G) (G) holds for any digraph G.

For an undirected graph G let us call the digraph we obtain from G by directing all its edges in both ways the symmetrically directed equivalent of G. In Shannon's own language the capacity (cf. [24]) of the complement of an (undirected) graph G can be defined as the Sperner capacity of its symmetrically directed equivalent. We denote this quantity by c(G) and by slight abuse of the terminology we also refer to it as the Shannon capacity of G whenever it may not cause confusion.

Thus Sperner capacity is a generalization of Shannon capacity. It is a true generalization in the sense that there exist digraphs the Sperner capacity of which is different from the Shannon capacity (c(G) value) of its underlying undirected graph. Denoting by G both an arbitrary digraph and its underlying undirected graph, it follows from the definitions that (G) c(G) always holds. The smallest example with strict inequality in the previous relation is a cyclically oriented triangle, cf. [8,6]. (See also [5] for an early and different attempt to generalize Shannon capacity to directed graphs.)

Shannon capacity is known to be a graph invariant that is difficult to determine (not only in the algorithmic but in any sense), and it is unknown for many relatively small and simple graphs, for example, for all odd cycles of length at least 7. This already shows that the more general invariant Sperner capacity cannot be easy to determine either. For a survey on graph invariants defined via powers, including Shannon and Sperner capacities, we refer the reader to [3]. There is an interesting and important connection between Sperner capacity and extremal set theory, introduced in [19] and fully explored in [15]. Several problems of this flavour are also discussed in [18].

Main result

Alon [1] proved that for any digraph G

where + (G) is the maximum out-degree of the graph G and similarly − (G) is the maximum in-degree. The proof relies on a linear algebraic method similar to the one already used in [6] for a special case (cf. also [11] for a strengthening and cf. [2] for a general setup for this method in case of undirected graphs). We also use this method for proving the following stronger result: For each vertex a = (a 1 , . . . , a n ) ∈ V (G n ) we define a polynomial

Let K be a symmetric clique in G n . If a ∈ K, b ∈ K then by definition P a,c (c(b 1 ), . . . , c(b n )) = 0 if b = a, while P a,c (c(a 1 ), . . . , c(a n )) = 0 by the properness of colouring c. This implies that the polynomials {P a,c (x 1 , . . . , x n )} a∈K are linearly independent over the reals. This can be shown in the usual way: substituting c(b) into a∈K a P a,c (x) = 0 we obtain b = 0 and this can be done for each b ∈ K. Since the degree of x i in P a,c (x) is at most d (G) − 1, the dimension of the linear space generated by our polynomials is bounded from above by [ d (G)] n . By the previous paragraph, this is also an upper bound for |K|. Choosing K to be a symmetric clique of maximum size we obtain s (G n ) [ d (G)] n and thus the statement.

Let G rev denote the "reverse of G", i.e., the digraph we obtain from G by reversing the direction of all of its edges. Since obviously (G) = (G rev ), Theorem 1 has the following trivial corollary:

In Sections 7 and 8 we will strengthen Theorem 1 by introducing a fractional version of d (G).

Application: odd cycles

We call an oriented cycle alternating if it has at most one vertex of outdegree 1. (In stating the following results we follow the convention that an oriented graph is a graph without oppositely directed edges between the same two points, while a general directed graph may contain such pairs of edges.) Clearly, in any oriented cycle the number of vertices of outdegree 2 equals the number of vertices of outdegree 0. Thus, in particular, an oriented odd cycle of length 2k + 1 is alternating if it has k points of outdegree zero, k points of outdegree 2 and only 1 point of outdegree 1. It takes an easy checking that up to isomorphism there is only one orientation of C 2k+1 which is alternating.

Theorem 2. Let G be an oriented odd cycle that is not alternating. Then

Proof. Since any digraph with at least one edge has Sperner capacity at least 2 (see the Remark after Definition 5), it is enough to prove that 2 is also an upper bound.

Colour the vertices of G so that two points receive the same colour if and only if they have a common in-neighbour, i.e., a vertex sending an oriented edge to both of them. It is easy to check that this colouring is proper if and only if the odd cycle G is not alternating. In this case, our colouring also has the property that any vertex has only one colour in its out-neigbourhood proving d (G) = 2. Then the statement follows by Theorem 1.

Remark. It is easy to see that the following slightly stronger version of the previous theorem can be proven similarly: If G is a directed odd cycle not containing an alternating odd cycle, then (G) = 2.

The Sperner capacity of an alternating odd cycle can indeed be larger than 2. This is obvious for C 3 , where the alternating orientation produces a transitive clique of size 3. A construction proving that the Sperner capacity of the alternating C 5 is at least √ 5 is given in [13], and this is further analyzed in [22]. The construction is clearly best possible by the celebrated result of Lovász [20] showing c(C 5 ) = √ 5. In [22] the invariant D(G) = max (G) was defined where the maximization is over all orientations of G. It follows from the definitions that D(G) c(G), and it is asked in [22] whether one always has equality. No counterexample is known, while equality is trivial if (G) = (G) (just orient a maximum size clique transitively) and it is proven for vertextransitive self-complementary graphs in [22]. Denoting the alternatingly oriented C 2k+1 by C alt 2k+1 Theorem 2 has the following immediate corollary:

holds for every positive integer k.

The discussion in this section becomes more relevant in the light of a recent result by Bohman and Holzman [7]. Until recently it was not known whether the Shannon capacity (in our complementary sense) of the odd cycle C 2k+1 , i.e., c(C 2k+1 ) is larger than 2 for any value of k > 2. In [7] an affirmative answer to this question was given by an ingenious construction, showing that this is always the case, i.e., c(C 2k+1 ) > 2 for every positive integer k. This means that the bound provided by d (G) goes beyond the obvious upper bound c(G) of Sperner capacity in case of non-alternatingly oriented odd cycles, i.e., the following consequence of Theorem 2 can also be formulated:

Corollary 3. If k is any positive integer and C →

2k+1 is a non-alternatingly oriented C 2k+1 , then

It is a natural idea to try to use the Bohman-Holzman construction for alternatingly oriented odd cycles and check whether the so-obtained sets of vertices inducing cliques in the appropriate power graphs will form transitive cliques in the oriented case. (If the answer were yes it would prove D(C 2k+1 ) > 2 for every k strengthening the result c(C 2k+1 ) > 2 of [7].) This idea turned out to work in the case of C 7 , thus showing D(C 7 ) > 2. (To record this we list the 17 vertices of C 4 7 that form a transitive clique defined by their ordering on this list. The labels of the vertices of C 7 are the first 7 non-negative integers as in [7] and the unique point with outdegree and indegree 1 is the point labelled 5. Here we give the vertices simply as sequences. Thus the list is: 4444, 0520, 2030, 2051, 0605, 1205, 1320, 3006, 3012, 5106, 5112, 0561, 0613, 1213, 6130, 6151, 1361.) Strangely, however, the same construction did not work for C 9 : after our unsuccessful attempts to prove a similar statement, Attila Sali wrote a computer program to check whether the clique of Bohman and Holzman in the 8th power of an alternating C 9 contains a transitive clique of the same size and the answer turned out to be negative. (Again, to record more than just this fact, we give six vertices of C 8 9 that form a directed cycle without inversely oriented edges in the clique of Bohman and Holzman whenever the path obtained after deleting vertex 5 of C 9 is oriented alternatingly. The existence of this cycle shows that the Bohman-Holzman clique does not contain a transitive clique whenever the only outdegree 1, indegree 1 point of the alternatingly oriented C 9 is 4 or 6 (or 5, but this case is less important), that is one of the neighbours of 5, the point the construction distinguishes. So the promised cycle is: 20302040, 12072040, 12140720, 40121207, 20401320, 07204012.) In spite of this, we believe that the Sperner capacity of alternating odd cycles will achieve the corresponding Shannon capacity value c(C 2k+1 ).

One more remark is in order. It is easy to check that the vertices of C alt 2k+3 can be mapped to those of C alt 2k+1 in an edge-preserving manner. This immediately implies that (C alt 2k+3 ) (C alt 2k+1 ), i.e., if there were any odd cycle C 2k+1 with D(C 2k+1 ) = 2, then the same must hold for all longer odd cycles as well.

The undirected case

Since identifying with any undirected graph G its symmetrically directed equivalent gives both (G) = c(G) and d (G) = (G), it is immediate from Theorem 1 that c(G) (G). We will show, however, that (G) is always bounded from below by the fractional chromatic number of G, which in turn is a well known upper bound for c(G), cf. [24,20]. Thus, unlike in the directed case, the local chromatic number does not give us new information about Shannon capacity. Looking at it from another perspective, this relation tells us something about the behaviour of the local chromatic number. (For more on this other perspective, see the follow up paper [25].)

One of the main tools in the investigations of the local chromatic number in [10] is the recognition of the relevance of the universal graphs U (m, k) defined as follows. (From now on we will use the notation [m] = {1, . . . , m}). Definition 6 (Erdős et al. [10]). Let the graph U (m, k) for positive integers k m be defined as follows: The relevance of these graphs is expressed by the following lemma. Recall that a homomorphism from some graph F to a graph G is an edge-preserving mapping of V (F ) to V (G). U (m, k). In particular, (G) k if and only if there exists an m such that G admits a homomorphism to U (m, k).

Lemma 1 (Erdős et al. [10]). A graph G admits a proper colouring c with m colours and max v∈V (G) |{c(u) : u ∈ G (v)}| k if and only if there exists a homomorphism of G to

We use these graphs to prove the relation between the fractional chromatic number and the local chromatic number.

Recall that the fractional chromatic number is * (G) = min A∈S(G) w(A) where S(G) denotes the family of independent sets of graph G and the minimization is over all nonnegative weightings w : S(G) → R satisfying A x w(A) 1 for every x ∈ V (G). It is straightforward from the definition that * (G) (G) holds for any graph G. Another important fact we will use is that if G is vertex-transitive, then * (G) = |V (G)| (G) . For a proof of this fact and for further information about the fractional chromatic number we refer to the books [23,16].

Theorem 3. For any graph G (G) * (G).

The proof relies on the following simple observation: gives an automorphism, and any vertex can be mapped to any other by such a permutation.) Consider those vertices (x, A) for which x a i for all a i ∈ A. These form an independent set S. Thinking about the vertices (x, A) as k-tuples with one distinguished element and the elements of S as those ktuples whose distinguished element is the smallest one, we immediately get * (U (m, k) In the rest of this section we formulate a consequence of Theorem 3 for the asymptotic behaviour of the local chromatic number with respect to the co-normal power of graphs.

It is a well-known theorem of McEliece and Posner [21] (cf. also Berge and Simonovits [4] and, for this particular formulation, [23]) that lim n→∞ n (G n ) = * (G).

It is equally well-known (cf., e.g., [23,Corollary 3.4.2]) that * (G n ) = [ * (G)] n . These two statements and Theorem 3 immediately imply the following:

where the last equality is by the McEliece-Posner theorem mentioned above.

Corollary 4 can be proved also in a different way using the techniques of [9]. This latter kind of proof can be generalized to show a similar statement for d (G).

Fractional colourings

Now we define the fractional version of the local chromatic number.

It is easy to see that d (G[K r ]) = d (G, r) for every digraph G and positive integer r.

It is also not difficult to see that s ((G[K r ]) n ) = r n s (G n ) for every n. Indeed, any vertex of G n can be substituted by r n vertices of (G[K r ]) n in the natural way and a symmetric clique K of G n becomes a symmetric clique of size

To see that equality holds let us denote by a(x) the unique vertex of G n from which x ∈ (G[K r ]) n can be obtained by the previous substitution. (Thus the set A x := {y : a(y) = a(x)} has r n elements for every x ∈

The crucial observation is that if K is a symmetric clique in (G[K r ]) n and x ∈ K, then K ∪ A x is still a symmetric clique (it may be identical to K but may also be larger). Thus maximal symmetric cliques of (G[K r ]) n can always be obtained as the union of some sets A x , which means that they can be obtained as "blown up" versions of symmetric cliques of G n . This proves our claim that s ((G[K r ]) n ) = r n s (G n ). This equality implies (G[K r ]) = r (G) for every digraph G and positive integer r.

The observations of the previous paragraph provide the following strengthening of Theorem 1.

Theorem 4. For every digraph G

Proof. By Theorem 1 and the previous observations we have

Since this holds for every r we can write

We can formulate again the following trivial corollary:

To illustrate the usefulness of Theorem 4 we consider the complement of a 7-cycle with its only orientation in which all triangles are oriented cyclically. We denote this graph by D 7 (abbreviating double 7-cycle). None of the earlier bounds we know give a better upper bound for the Sperner capacity of D 7 than 3. Now we can improve on this. This example can be further generalized as follows. Let D 2k+1 denote the following oriented graph:

Observe that this definition is consistent with the earlier definition of D 7 and that the underlying undirected graph of D 2k+1 is the complement of the odd cycle C 2k+1 . Now we can state the following:

In particular, (D 2k+1 ) = k 2 + 1 if k is even.

Proof. It is easy to verify for the transitive clique number that tr(D 2k+1 )= k−1 2 +1 and this gives the lower bound. The upper bound is proven by assigning weight 1 2 to every 2-element independent set of D 2k+1 which clearly gives a fractional colouring. The weight thus assigned to any closed out-neighbourhood is k 2 + 1 giving the upper bound by Theorem 4.

If k is even, the two bounds coincide.

We remark that while the upper bound in Proposition 2 generalizes that of Proposition 1, the lower bound does not; it is weaker in case k = 3 than that of Proposition 1. Therefore we consider the oriented graph D 7 a particularly interesting instance of the problem.

As it was the case without fractionalization, Theorem 4 does not give us new information in the undirected case, i.e., about Shannon capacity. The reason for this is the following relation:

Theorem 5. Let G be an undirected graph. Then * (G) = * (G).

To prove Theorem 5 we need the following generalization of the universal graphs U (m, k): Proof. The proof is more or less identical to that of Lemma 1 (cf. [10]). If the required colouring f exists then assign to each vertex v a pair of sets of colours (X, A) with X = f (v) and {u,v}∈E(G) f (u) ⊆ A. If f has the required properties then this assignment is indeed a homomorphism to U r (m, k).

On the other hand, if the required homomorphism h exists then the r-fold colouring f defined by the X-part of h(v) = (X, A) as f (v) satisfies the requirements.

The following lemma is a generalization of Lemma 2: vertices (X i , A i ) with the property that if i = j then X i ∩ X j = ∅. If X i ∪ A i = X j ∪ A j , then the latter is the very same condition as non-adjacency in U r (m, k). Thus we know (U r (m, k) We note that universal graphs can also be defined for the directed version of the local chromatic number. Denoting these graphs by

To show the analog of Lemma 1 is straightforward. Comparing U d (m, k) to U (m, k) one can see that the symmetrically directed edges of U d (m, k) are exactly the (undirected) edges present in U (m, k). This means (but the same can be seen also directly) that s (U d (m, k)

Fractional covers

A non-negative real-valued function g :

The most general upper bound on (G) we prove in this paper is given by the following inequality that generalizes Theorem 4 along the lines of a result (Theorem 2) of [11].

Theorem 6. For any digraph G we have

where the minimization is over all fractional covers g of V (G) and G[U ] denotes the digraph induced by G on U ⊆ V (G).

By (G) = (G rev ) we again have the following immediate corollary (cf. Corollary 1 of Theorem 1).

The proof of Theorem 6 is almost identical to that of Theorem 2 of [11]. Yet, we give the details for the sake of completeness.

We need some lemmas. Following [2], we can speak about the representation of a (di)graph G = (V , E) over a subspace F of polynomials in m variables over a field F . Such a representation is an assignment of a polynomial f v in F and a vector a v ∈ F m to each vertex v ∈ V such that the following two conditions hold:

Notice that we adapted the description of a representation given in [2] to our terminology (where capacities are defined via cliques instead of stable sets) and to digraphs.

The following two lemmas are from [2]. Their proofs are essentially identical to those of Lemma 2.2 and Lemma 2.3 in [2] (after some trivial changes caused by the different language). Our next lemma is analogous to Proposition 1 of [11]. Lemma 7. Let F 1 , F 2 , . . . , F n be digraphs. Then

Proof. First observe that the argument for s ((G[K r ]) n ) = r n s (G n ) that led us to state Theorem 4 generalizes to

Each summand in this last inequality satisfies by Lemma 7

Substituting this into the previous inequality we get

where the summations are meant with multiplicities.

Since K is a maximum size symmetric clique of G n and the multiplicity of U i in U is h(U i ), we obtained

This implies

where the minimization is over all fractional covers g of V (G), i.e., we arrived at the statement.

To illustrate that the bound of Theorem 6 may indeed give an improvement over that of Theorem 4 (or, in fact, over that of Corollary 5) consider the following digraph G. Let V (G) = {1, 2, . . . , 2k+1, a, b} and E(G)=E(C → 2k+1 )∪{(a, i), (i, b) : i ∈ {1, . . . , 2k+1}}, where C → 2k+1 is an arbitrary non-alternatingly oriented cycle on 2k + 1 vertices. It is easy to check that * d (G) = 3+ 1 k (and also * d (G rev ) = 3+ 1 k ) , i.e., Theorem 4 gives (G) 3+ 1 k only, while Theorem 6 gives (G) 3. Indeed, using the fractional cover (which is also an integer cover) g(V 1 ) = g(V 2 ) = 1, where V 1 = {1, 2, . . . , 2k + 1}, V 2 = {a, b} (and g(U ) = 0 for all other U ⊆ V (G)) we get (G) *

. This bound is sharp since G contains transitive triangles.