Cyclic cycle systems of the complete multipartite graph

In this paper, we study the existence problem for cyclic $\ell$-cycle decompositions of the graph $K_m[n]$, the complete multipartite graph with $m$ parts of size $n$, and give necessary and sufficient conditions for their existence in the case that $2\ell \mid (m-1)n$

Resolution of the Oberwolfach problem

The Oberwolfach problem, posed by Ringel in 1967, asks for a decomposition of $K_{2n+1}$ into edge-disjoint copies of a given $2$-factor. We show that this can be achieved for all large $n$. We actually prove a significantly more general result, which allows for decompositions into more general types of factors. In particular, this also resolves the Hamilton-Waterloo problem for large $n$.Comment: 28 page

Decomposition of Certain Complete Graphs and Complete Multipartite Graphs into Almost-bipartite Graphs and Bipartite Graphs

In his classical paper [14], Rosa introduced a hierarchical series of labelings called ρ, σ, β and α labeling as a tool to settle Ringel’s Conjecture which states that if T is any tree with m edges then the complete graph K2m+1 can be decomposed into 2m + 1 copies of T . Inspired by the result of Rosa [14] many researchers significantly contributed to the theory of graph decomposition using graph labeling. In this direction, in 2004, Blinco et al. [6] introduced γ-labeling as a stronger version of ρ-labeling. A function g defined on the vertex set of a graph G with n edges is called a γ-labeling if (i) g is a ρ-labeling of G, (ii) G is a tripartite graph with vertex tripartition (A, B, C) with C = {c} and ¯b ∈ B such that {¯b, c} is the unique edge joining an element of B to c, (iii) g(a) \u3c g(v) for every edge {a, v} ∈ E(G) where a ∈ A, (iv) g(c) - g(¯b) = n. Further, Blinco et al. [6] proved a significant result that the complete graph K2cn+1 can be cyclically decomposed into c(2cn + 1) copies of any γ-labeled graph with n edges, where c is any positive integer. Recently, in 2013, Anita Pasotti [4] introduced a generalisation of graceful labeling called d-divisible graceful labeling as a tool to obtain cyclic G-decompositions in complete multipartite graphs. Let G be a graph of size e = d . m. A d-divisible graceful labeling of the graph G is an injective function g : V (G) → {0, 1, 2, . . . , d(m + 1) - 1} such that {|g(u) - g(v)|/{u, v} ∈ E(G)} = {1, 2, . . . , d(m + 1) - 1}\{m + 1, 2(m + 1), . . . , (d - 1)(m + 1)}. A d-divisible graceful labeling of a bipartite graph G is called as a d-divisible α-labeling of G if the maximum value of one of the two bipartite sets is less than the minimum value of the other one. Further, Anita Pasotti [4] proved a significant result that the complete multipartite graph K (e/d +1)×2dc can be cyclically decomposed into copies of d-divisible α-labeled graph G, where e is the size of the graph G and c is any positive integer (K (e/d +1)×2dc contains e/d + 1 parts each of size 2dc). Motivated by the results of Blinco et al. [6] and Anita Pasotti [4], in this paper we prove the following results. i) For t ≥ 2, disjoint union of t copies of the complete bipartite graph Km,n, where m≥ 3, n ≥ 4 plus an edge admits γ-labeling. ii) For t ≥ 2, t-levels shadow graph of the path Pdn+1 admits d-divisible α-labeling for any admissible d and n ≥ 1. Further, we discuss related open problems

Star Decompositions of Bipartite Graphs

In Chapter 1, we will introduce the definitions and the notations used throughout this thesis. We will also survey some prior research pertaining to graph decompositions, with special emphasis on star-decompositions and decompositions of bipartite graphs. Here we will also introduce some basic algorithms and lemmas that are used in this thesis. In Chapter 2, we will focus primarily on decomposition of complete bipartite graphs. We will also cover the necessary and sufficient conditions for the decomposition of complete bipartite graphs minus a 1-factor, also known as crown graphs and show that all complete bipartite graphs and crown graphs have a decomposition into stars when certain necessary conditions for the decomposition are met. This is an extension of the results given in "On claw-decomposition of complete graphs and complete bigraphs" by Yamamoto, et. al. We will propose a construction for the decomposition of the graphs. In Chapter 3, we focus on the decomposition of complete equipartite tripartite graphs. This result is similar to the results of "On Claw-decomposition of complete multipartite graphs" by Ushio and Yamamoto. Our proof is again by construction and we propose how it might extend to equipartite multipartite graphs. We will also discuss the 3-star decomposition of complete tripartite graphs. In Chapter 4 , we will discuss the star decomposition of 4-regular bipartite graphs, with particular emphasis on the decomposition of 4-regular bipartite graphs into 3-stars. We will propose methods to extend our strategies to model the problem as an optimization problem. We will also look into the probabilistic method discussed in "Tree decomposition of Graphs" by Yuster and how we might modify the results of this paper to star decompositions of bipartite graphs. In Chapter 5, we summarize the findings in this thesis, and discuss the future work and research in star decompositions of bipartite and multipartite graphs

About the Dedekind psi function in Pauli graphs

We study the commutation structure within the Pauli groups built on all decompositions of a given Hilbert space dimension $q$, containing a square, into its factors. The simplest illustrative examples are the quartit ($q=4$) and two-qubit ($q=2^2$) systems. It is shown how the sum of divisor function $\sigma(q)$ and the Dedekind psi function $\psi(q)=q \prod_{p|q} (1+1/p)$ enter into the theory for counting the number of maximal commuting sets of the qudit system. In the case of a multiple qudit system (with $q=p^m$ and $p$ a prime), the arithmetical functions $\sigma(p^{2n-1})$ and $\psi(p^{2n-1})$ count the cardinality of the symplectic polar space $W_{2n-1}(p)$ that endows the commutation structure and its punctured counterpart, respectively. Symmetry properties of the Pauli graphs attached to these structures are investigated in detail and several illustrative examples are provided.Comment: Proceedings of Quantum Optics V, Cozumel to appear in Revista Mexicana de Fisic

Decompositions of Triangle-Dense Graphs

High triangle density -- the graph property stating that a constant fraction of two-hop paths belong to a triangle -- is a common signature of social networks. This paper studies triangle-dense graphs from a structural perspective. We prove constructively that significant portions of a triangle-dense graph are contained in a disjoint union of dense, radius 2 subgraphs. This result quantifies the extent to which triangle-dense graphs resemble unions of cliques. We also show that our algorithm recovers planted clusterings in approximation-stable k-median instances.Comment: 20 pages. Version 1->2: Minor edits. 2->3: Strengthened {\S}3.5, removed appendi

Pauli graphs when the Hilbert space dimension contains a square: why the Dedekind psi function ?

We study the commutation relations within the Pauli groups built on all decompositions of a given Hilbert space dimension $q$, containing a square, into its factors. Illustrative low dimensional examples are the quartit ($q=4$) and two-qubit ($q=2^2$) systems, the octit ($q=8$), qubit/quartit ($q=2\times 4$) and three-qubit ($q=2^3$) systems, and so on. In the single qudit case, e.g. $q=4,8,12,...$, one defines a bijection between the $\sigma (q)$ maximal commuting sets [with $\sigma[q)$ the sum of divisors of $q$] of Pauli observables and the maximal submodules of the modular ring $\mathbb{Z}_q^2$, that arrange into the projective line $P_1(\mathbb{Z}_q)$ and a independent set of size $\sigma (q)-\psi(q)$ [with $\psi(q)$ the Dedekind psi function]. In the multiple qudit case, e.g. $q=2^2, 2^3, 3^2,...$, the Pauli graphs rely on symplectic polar spaces such as the generalized quadrangles GQ(2,2) (if $q=2^2$) and GQ(3,3) (if $q=3^2$). More precisely, in dimension $p^n$ ($p$ a prime) of the Hilbert space, the observables of the Pauli group (modulo the center) are seen as the elements of the $2n$-dimensional vector space over the field $\mathbb{F}_p$. In this space, one makes use of the commutator to define a symplectic polar space $W_{2n-1}(p)$ of cardinality $\sigma(p^{2n-1})$, that encodes the maximal commuting sets of the Pauli group by its totally isotropic subspaces. Building blocks of $W_{2n-1}(p)$ are punctured polar spaces (i.e. a observable and all maximum cliques passing to it are removed) of size given by the Dedekind psi function $\psi(p^{2n-1})$. For multiple qudit mixtures (e.g. qubit/quartit, qubit/octit and so on), one finds multiple copies of polar spaces, ponctured polar spaces, hypercube geometries and other intricate structures. Such structures play a role in the science of quantum information.Comment: 18 pages, version submiited to J. Phys. A: Math. Theo