68 research outputs found
Algorithmic and explicit determination of the Lovász number for certain circulant graphs
AbstractWe consider the problem of computing the Lovász theta function for circulant graphs Cn,J of degree four with n vertices and chord length J, 2⩽J⩽n. We present an algorithm that takes O(J) operations if J is an odd number, and O(n/J) operations if J is even. On the considered class of graphs our algorithm strongly outperforms the known algorithms for theta function computation. We also provide explicit formulas for the important special cases J=2 and J=3
On Hamilton decompositions of infinite circulant graphs
The natural infinite analogue of a (finite) Hamilton cycle is a two-way-infinite Hamilton path (connected spanning 2-valent subgraph).
Although it is known that every connected 2k-valent infinite circulant graph has a two-way-infinite Hamilton path, there exist many such graphs that do not have a decomposition into k edge-disjoint two-way-infinite Hamilton paths. This contrasts with the finite case where it is conjectured that every 2k-valent connected circulant graph has a decomposition into k edge-disjoint Hamilton cycles. We settle the problem of decomposing 2k-valent infinite circulant graphs into k edge-disjoint two-way-infinite Hamilton paths for k=2, in many cases when k=3, and in many other cases including where the connection set is ±{1,2,...,k} or ±{1,2,...,k - 1, 1,2,...,k + 1}
The (Generalized) Orthogonality Dimension of (Generalized) Kneser Graphs: Bounds and Applications
The orthogonality dimension of a graph over a field is
the smallest integer for which there exists an assignment of a vector with to every vertex , such that whenever and are
adjacent vertices in . The study of the orthogonality dimension of graphs is
motivated by various application in information theory and in theoretical
computer science. The contribution of the present work is two-folded.
First, we prove that there exists a constant such that for every
sufficiently large integer , it is -hard to decide whether the
orthogonality dimension of an input graph over is at most or
at least . At the heart of the proof lies a geometric result, which
might be of independent interest, on a generalization of the orthogonality
dimension parameter for the family of Kneser graphs, analogously to a
long-standing conjecture of Stahl (J. Comb. Theo. Ser. B, 1976).
Second, we study the smallest possible orthogonality dimension over finite
fields of the complement of graphs that do not contain certain fixed subgraphs.
In particular, we provide an explicit construction of triangle-free -vertex
graphs whose complement has orthogonality dimension over the binary field at
most for some constant . Our results involve
constructions from the family of generalized Kneser graphs and they are
motivated by the rigidity approach to circuit lower bounds. We use them to
answer a couple of questions raised by Codenotti, Pudl\'{a}k, and Resta (Theor.
Comput. Sci., 2000), and in particular, to disprove their Odd Alternating Cycle
Conjecture over every finite field.Comment: 19 page
On densities of lattice arrangements intersecting every i-dimensional affine subspace
In 1978, Makai Jr. established a remarkable connection between the
volume-product of a convex body, its maximal lattice packing density and the
minimal density of a lattice arrangement of its polar body intersecting every
affine hyperplane. Consequently, he formulated a conjecture that can be seen as
a dual analog of Minkowski's fundamental theorem, and which is strongly linked
to the well-known Mahler-conjecture.
Based on the covering minima of Kannan & Lov\'asz and a problem posed by
Fejes T\'oth, we arrange Makai Jr.'s conjecture into a wider context and
investigate densities of lattice arrangements of convex bodies intersecting
every i-dimensional affine subspace. Then it becomes natural also to formulate
and study a dual analog to Minkowski's second fundamental theorem. As our main
results, we derive meaningful asymptotic lower bounds for the densities of such
arrangements, and furthermore, we solve the problems exactly for the special,
yet important, class of unconditional convex bodies.Comment: 19 page
Hamilton decompositions of 6-regular abelian Cayley graphs
In 1969, Lovasz asked whether every connected, vertex-transitive graph has a Hamilton path. This question has generated a considerable amount of interest, yet remains vastly open. To date, there exist no known connected, vertex-transitive graph that does not possess a Hamilton path. For the Cayley graphs, a subclass of vertex-transitive graphs, the following conjecture was made:
Weak Lovász Conjecture: Every nontrivial, finite, connected Cayley graph is hamiltonian.
The Chen-Quimpo Theorem proves that Cayley graphs on abelian groups flourish with Hamilton cycles, thus prompting Alspach to make the following conjecture:
Alspach Conjecture: Every 2k-regular, connected Cayley graph on a finite abelian group has a Hamilton decomposition.
Alspach’s conjecture is true for k = 1 and 2, but even the case k = 3 is still open. It is this case that this thesis addresses.
Chapters 1–3 give introductory material and past work on the conjecture. Chapter 3 investigates the relationship between 6-regular Cayley graphs and associated quotient graphs. A proof of Alspach’s conjecture is given for the odd order case when k = 3. Chapter 4 provides a proof of the conjecture for even order graphs with 3-element connection sets that have an element generating a subgroup of index 2, and having a linear dependency among the other generators.
Chapter 5 shows that if Γ = Cay(A, {s1, s2, s3}) is a connected, 6-regular, abelian Cayley graph of even order, and for some1 ≤ i ≤ 3, Δi = Cay(A/(si), {sj1 , sj2}) is 4-regular, and Δi ≄ Cay(ℤ3, {1, 1}), then Γ has a Hamilton decomposition. Alternatively stated, if Γ = Cay(A, S) is a connected, 6-regular, abelian Cayley graph of even order, then Γ has a Hamilton decomposition if S has no involutions, and for some s ∈ S, Cay(A/(s), S) is 4-regular, and of order at least 4.
Finally, the Appendices give computational data resulting from C and MAGMA programs used to generate Hamilton decompositions of certain non-isomorphic Cayley graphs on low order abelian groups
Clique-circulants and the stable set polytope of fuzzy circular interval graphs
In this paper, we give a complete and explicit description of the rank facets of the stable set polytope for a class of claw-free graphs, recently introduced by Chudnovsky and Seymour (Proceedings of the Bristish Combinatorial Conference, 2005), called fuzzy circular interval graphs. The result builds upon the characterization of minimal rank facets for claw-free graphs by Galluccio and Sassano (J. Combinatorial Theory 69:1-38, 2005) and upon the introduction of a superclass of circulant graphs that are called clique-circulants. The new class of graphs is invetigated in depth. We characterize which clique-circulants C are facet producing, i.e. are such that Sigma upsilon epsilon V(C) chi(upsilon) <= alpha(C) is a facet of STAB(C), thus extending a result of Trotter (Discrete Math. 12:373-388, 1975) for circulants. We show that a simple clique family inequality (Oriolo, Discrete Appl. Math. 132(2):185-201, 2004) may be associated with each clique-circulant C subset of G, when G is fuzzy circular interval. We show that these inequalities provide all the rank facets of STAB(G), if G is a fuzzy circular interval graph. Moreover we conjecture that, in this case, they also provide all the non-rank facets of STAB(G) and offer evidences for this conjecture
On Minrank and Forbidden Subgraphs
The minrank over a field of a graph on the vertex set
is the minimum possible rank of a matrix such that for every , and
for every distinct non-adjacent vertices and in . For an
integer , a graph , and a field , let
denote the maximum possible minrank over of an -vertex graph
whose complement contains no copy of . In this paper we study this quantity
for various graphs and fields . For finite fields, we prove by
a probabilistic argument a general lower bound on , which
yields a nearly tight bound of for the triangle
. For the real field, we prove by an explicit construction that for
every non-bipartite graph , for some
. As a by-product of this construction, we disprove a
conjecture of Codenotti, Pudl\'ak, and Resta. The results are motivated by
questions in information theory, circuit complexity, and geometry.Comment: 15 page
Information transfer fidelity in spin networks and ring-based quantum routers
Spin networks are endowed with an Information Transfer Fidelity (ITF), which defines an absolute upper bound on the probability of transmission of an excitation from one spin to another. The ITF is easily computable but the bound can be reached asymptotically in time only under certain conditions. General conditions for attainability of the bound are established and the process of achieving the maximum transfer probability is given a dynamical model, the translation on the torus. The time to reach the maximum probability is estimated using the simultaneous Diophantine approximation, implemented using a variant of the Lenstra-Lenstra-Lov\'asz (LLL) algorithm. For a ring with uniform couplings, the network can be made a metric space by defining a distance (satisfying the triangle inequality) that quantifies the lack of transmission fidelity between two nodes. It is shown that transfer fidelities and transfer times can be improved significantly by means of simple controls taking the form of non-dynamic, spatially localized bias fields, opening up the possibility for intelligent design of spin networks and dynamic routing of information encoded in them, while being more flexible than engineering fixed couplings to favor some transfers, and less demanding than control schemes requiring fast dynamic controls
- …