is the disjoint union ofGand H. This equality immediately follows from Corollary 8 since the analogous statement, a∗(G]H) = max{a∗(G), a∗(H)} is straightforward. In [11] it is shown thatA(G]H) =A(G×H), therefore we get the following result.
Corollary 9. For every two graphsG and H we have
A(G]H) =A(G×H) = max{A(G), A(H)}.
The authors of [15] also addressed the question whetherA(G) is computable, and if so what is its complexity. They showed that in the case when Gis bipartite then A(G) = 12 ifG has a perfect matching, andA(G) = 1 otherwise. Hence for bipartite graphsA(G) can be determined in polynomial time. Furthermore, it is proven in [11] thata(G)≤ 12 if and only if Gcontains a fractional perfect matching. Therefore given an input graphG, determining whether A(G) = 1 or A(G) ≤ 12 can be done in polynomial time. From Corollary 8 we can conclude that the problem of deciding whether A(G) > t for a given graph G and a given value t, is in NP.
Moreover it is not hard to prove that it is in fact NP-complete. (The maximum independent set problem has a Karp-reduction to this problem, by adding sufficiently many vertices to the graph which are connected to each other and every other vertex of the graph, and choosing t appropriately.)
Any rational number in (0,12]∪ {1} is the ultimate categorical independence ratio for some graphG, as it is shown in [15]. Here we remark that we obtained thatA(G) cannot be irrational, solving another problem mentioned in [15].
As a consequence of Corollary 8 we also have the following characterization of self-universal graphs. We call a graph empty if it has no edge. For every other graphGit holds thati(G)<1.
20
2 The ultimate categorical independence ratio
Corollary 10. A non-empty graph Gis self-universal if and only ifa(G) =i(G)andi(G)≤ 12. In other words, a nonempty graph G is self-universal iff the expression |U|+||NU|
G(U)| reach its maximum (also) for maximum-sized independent sets among all independent sets of Gand this maximum is at most 12. Clearly, from this result Theorem 3 of Section 2.2 also follows.
Chapter 3
The asymptotic value of the Hall-ratio for categorical and lexicographic power
The Hall-ratio of a graph Gwas investigated in [20, 21] where it is defined as ρ(G) = max
|V(H)|
α(H) : H ⊆G
,
that is, as the ratio of the number of vertices and the independence number maximized over all subgraphs ofG. (See also [23] and some of the references therein for an earlier appearance of the same notion on a different name.) The asymptotic values of the Hall-ratio for different graph powers were investigated by Simonyi [49]. He considered the (appropriately normalized) asymp-totic values of the Hall-ratio for the exponentiations called normal, co-normal, lexicographic and categorical, respectively.
All the above four graph powers of the graph Gare defined on the k-length sequences over V(G). In the normal powerGktwo sequences are adjacent iff their elements at every coordinate are either equal or form an edge in G. In the co-normal power Gk two such sequences are connected iff there is some coordinate where the corresponding elements of the two sequences form an edge ofG. The asymptotic value of the Hall-ratio with respect to the co-normal power is defined ash(G) = lim
k→∞
pk
ρ(Gk), the analogous asymptotic value for the normal power is denoted by h(G). Simonyi [49] proved that h(G) = χf(G), where χf(G) is the fractional chromatic number of graph G, while h(G) = R(G), where R(G) denotes the so-called Witsenhausen rate. The latter is the normalized asymptotic value of the chromatic number with respect to the normal power and is introduced by Witsenhausen in [52] where its information theoretic relevance is also explained. Thefractional chromatic number is the well-known graph invariant one obtains from the fractional relaxation of the integer program defining the chromatic number.
3.1
S(G) denotes the set of the independent sets ofG.
In the lexicographic powerG◦ktwo sequences of the original vertices are adjacent iff they are adjacent in the first coordinate where they differ. The ultimate lexicographic Hall-ratio of graph Gish◦(G) = lim
k→∞
pk
ρ(G◦k). In the categorical powerG×ktwo sequences of the original vertices are connected iff their elements form an edge inGat every coordinate. The ultimate categorical Hall-ratio of graph G is h×(G) = lim
k→∞ρ(G×k). (Note, that we do not need any normalization here.) Simonyi [49] conjectured that also for the lexicographic power and for the categorical power we get the fractional chromatic number as the asymptotic value. In this chapter we prove both of his conjectures. In Section 3.1 and 3.2 the ultimate lexicographic Hall-ratio, in Section 3.3, 3.4 and 3.5 the ultimate categorical Hall-ratio is discussed.
In the proofs we will also need the fractional relaxation of the clique number. A func-tion g : V(G) → [0,1] for which ∀U ∈ S(G) : P g is a fractional clique ofGwith value z(g)}. A fractional clique of G is called optimal if its value is ωf(G). (Figure 9 illustrates a fractional colouring and a fractional clique of a graph.) The duality theorem of linear programming implies that χf(G) = ωf(G) for every graph G.
See [48] for more details.
Figure 9: An optimal fractional colouring and an optimal fractional clique of a wheel graph.
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3 The asymptotic value of the Hall-ratio for categorical and lexicographic power
3.1 The ultimate lexicographic Hall-ratio
For two graphsF andG, theirlexicographic product F◦Gis defined on the vertex setV(F◦G) = V(F)× V(G) with edge set E(F ◦ G) = {{(u1, v1),(u2, v2)} : {u1, u2} ∈ E(F), oru1 = u2 and {v1, v2} ∈E(G)}. The lexicographic product F◦Gis also known as the substitution of G into all vertices of F, the name we use follows the book [39]. The nth lexicographic power G◦n is then-fold lexicographic product ofG. That is, the lexicographic power is defined on the vertex sequences of the original graph and we connect two such sequences iff they are adjacent in the first coordinate where they differ. (See Figure 10.)
G
G◦2
Figure 10: The lexicographic square of a graph. (Double lines mean that the corresponding vertex classes are totally connected.)
Definition ([49]). Theultimate lexicographic Hall-ratio of graphGis h◦(G) = lim
n→∞
pn
ρ(G◦n).
The normal and co-normal products of two graphsF andGare also defined onV(F)×V(G) as vertex sets and their edge sets are such that E(F G) ⊆ E(F ◦G) ⊆ E(F ·G) holds, where F G denotes the normal, F ·G the co-normal product of F and G. (In particular, {(u1, v1),(u2, v2)} ∈ E(F G) if {u1, u2} ∈ E(F) and {v1, v2} ∈ E(G), or {u1, u2} ∈ E(F) and v1 =v2, oru1 =u2 and {v1, v2} ∈E(G), while {(u1, v1),(u2, v2)} ∈E(F ·G) if{u1, u2} ∈ E(F) or {v1, v2} ∈E(G).)
As we have seen at the beginning of this chapter, denoting by h(G) and h(G) the normalized asymptotic values analogous toh◦(G) for the normal and co-normal power, respectively, Simonyi [49] proved that h(G) = χf(G), where χf(G) is the fractional chromatic number of graph G, while h(G) =R(G), where R(G) denotes the Witsenhausen rate.
It follows from the above discussion that the value ofh◦(G) falls into the interval [R(G), χf(G)].
We remark that the lower bound R(G) is sometimes better but sometimes worse than the easy