290 research outputs found

    The Expressive Power of CSP-Quantifiers

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    A generalized quantifier QK is called a CSP-quantifier if its defining class K consists of all structures that can be homomorphically mapped to a fixed finite template structure. For all positive integers n ≥ 2 and k, we define a pebble game that characterizes equivalence of structures with respect to the logic Lk∞ω(CSP+n ), where CSP+n is the union of the class Q1 of all unary quantifiers and the class CSPn of all CSP-quantifiers with template structures that have at most n elements. Using these games we prove that for every n ≥ 2 there exists a CSP-quantifier with template of size n + 1 which is not definable in Lω∞ω(CSP+n ). The proof of this result is based on a new variation of the well-known Cai-Fürer-Immerman construction.publishedVersionPeer reviewe

    The long-range Falicov-Kimball model and the amorphous Kitaev model: Quantum many-body systems I have known and loved

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    Large systems of interacting objects can give rise to a rich array of emergent behaviours. Make those objects quantum and the possibilities only expand. Interacting quantum many-body systems, as such systems are called, include essentially all physical systems. Luckily, we don't usually need to consider this full quantum many-body description. The world at the human scale is essentially classical (not quantum), while at the microscopic scale of condensed matter physics we can often get by without interactions. Strongly correlated materials, however, do require the full description. Some of the most exciting topics in modern condensed matter fall under this umbrella: the spin liquids, the fractional quantum Hall effect, high temperature superconductivity and much more. Unfortunately, strongly correlated materials are notoriously difficult to study, defying many of the established theoretical techniques within the field. Enter exactly solvable models, these are interacting quantum many-body systems with extensively many local symmetries. The symmetries give rise to conserved charges. These charges break the model up into many non-interacting quantum systems which are more amenable to standard theoretical techniques. This thesis will focus on two such exactly solvable models. The first, the Falicov-Kimball (FK) model is an exactly solvable limit of the famous Hubbard model which describes itinerant fermions interacting with a classical Ising background field. Originally introduced to explain metal-insulator transitions, it has a rich set of ground state and thermodynamic phases. Disorder or interactions can turn metals into insulators and the FK model features both transitions. We will define a generalised FK model in 1D with long-range interactions. This model shows a similarly rich phase diagram to its higher dimensional cousins. We use an exact Markov Chain Monte Carlo method to map the phase diagram and compute the energy resolved localisation properties of the fermions. This allows us to look at how the move to 1D affects the physics of the model. We show that the model can be understood by comparison to a simpler model of fermions coupled to binary disorder. The second, the Kitaev Honeycomb (KH) model, was the one of the first solvable 2D models with a Quantum Spin Liquid (QSL) ground state. QSLs are generally expected to arise from Mott insulators, when frustration prevents magnetic ordering all the way to zero temperature. The QSL state defies the traditional Landau-Ginzburg-Wilson paradigm of phases being defined by local order parameters. It is instead a topologically ordered phase. Recent work generalising non-interacting topological insulator phases to amorphous lattices raises the question of whether interacting phases like the QSLs can be similarly generalised. We extend the KH model to random lattices with fixed coordination number three generated by Voronoi partitions of the plane. We show that this model remains solvable and hosts a chiral amorphous QSL ground state. The presence of plaquettes with an odd number of sides leads to a spontaneous breaking of time reversal symmetry. We unearth a rich phase diagram displaying Abelian as well as a non-Abelian QSL phases with a remarkably simple ground state flux pattern. Furthermore, we show that the system undergoes a phase transition to a conducting thermal metal state and discuss possible experimental realisations.Open Acces

    A guide to the Rado graph : exploring structural and logical properties of the Rado graph

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    Dissertation (MSc (Mathematics))--University of Pretoria, 2023.The Rado graph, denoted R, is the unique (up to isomorphism) countably infinite random graph. It satisfies the extension property, that is, for two finite sets U and V of vertices of R there is a vertex outside of both U and V connected to every vertex in U and none in V . This property of the Rado graph allows us to prove quite a number of interesting results, such as a 0-1-law for graphs. Amongst other things, the Rado graph is partition regular, non-fractal, ultrahomogeneous, saturated, resplendent, the Fra´ıss´e-limit of the class of finite graphs, a non-standard model of the first-order theory of finite graphs, and has a complete decidable theory. We classify the definable subgraphs of the Rado graph and prove results for finite graphs that satisfy a restricted version of the extension property. We also mention some parallels between the rationals viewed as a linear order and the Rado graph.Mathematics and Applied MathematicsMSc (Mathematics)Unrestricte

    On learning the structure of clusters in graphs

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    Graph clustering is a fundamental problem in unsupervised learning, with numerous applications in computer science and in analysing real-world data. In many real-world applications, we find that the clusters have a significant high-level structure. This is often overlooked in the design and analysis of graph clustering algorithms which make strong simplifying assumptions about the structure of the graph. This thesis addresses the natural question of whether the structure of clusters can be learned efficiently and describes four new algorithmic results for learning such structure in graphs and hypergraphs. The first part of the thesis studies the classical spectral clustering algorithm, and presents a tighter analysis on its performance. This result explains why it works under a much weaker and more natural condition than the ones studied in the literature, and helps to close the gap between the theoretical guarantees of the spectral clustering algorithm and its excellent empirical performance. The second part of the thesis builds on the theoretical guarantees of the previous part and shows that, when the clusters of the underlying graph have certain structures, spectral clustering with fewer than k eigenvectors is able to produce better output than classical spectral clustering in which k eigenvectors are employed, where k is the number of clusters. This presents the first work that discusses and analyses the performance of spectral clustering with fewer than k eigenvectors, and shows that general structures of clusters can be learned with spectral methods. The third part of the thesis considers efficient learning of the structure of clusters with local algorithms, whose runtime depends only on the size of the target clusters and is independent of the underlying input graph. While the objective of classical local clustering algorithms is to find a cluster which is sparsely connected to the rest of the graph, this part of the thesis presents a local algorithm that finds a pair of clusters which are densely connected to each other. This result demonstrates that certain structures of clusters can be learned efficiently in the local setting, even in the massive graphs which are ubiquitous in real-world applications. The final part of the thesis studies the problem of learning densely connected clusters in hypergraphs. The developed algorithm is based on a new heat diffusion process, whose analysis extends a sequence of recent work on the spectral theory of hypergraphs. It allows the structure of clusters to be learned in datasets modelling higher-order relations of objects and can be applied to efficiently analyse many complex datasets occurring in practice. All of the presented theoretical results are further extensively evaluated on both synthetic and real-word datasets of different domains, including image classification and segmentation, migration networks, co-authorship networks, and natural language processing. These experimental results demonstrate that the newly developed algorithms are practical, effective, and immediately applicable for learning the structure of clusters in real-world data

    Subchromatic numbers of powers of graphs with excluded minors

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    A kk-subcolouring of a graph GG is a function f:V(G){0,,k1}f:V(G) \to \{0,\ldots,k-1\} such that the set of vertices coloured ii induce a disjoint union of cliques. The subchromatic number, χsub(G)\chi_{\textrm{sub}}(G), is the minimum kk such that GG admits a kk-subcolouring. Ne\v{s}et\v{r}il, Ossona de Mendez, Pilipczuk, and Zhu (2020), recently raised the problem of finding tight upper bounds for χsub(G2)\chi_{\textrm{sub}}(G^2) when GG is planar. We show that χsub(G2)43\chi_{\textrm{sub}}(G^2)\le 43 when GG is planar, improving their bound of 135. We give even better bounds when the planar graph GG has larger girth. Moreover, we show that χsub(G3)95\chi_{\textrm{sub}}(G^{3})\le 95, improving the previous bound of 364. For these we adapt some recent techniques of Almulhim and Kierstead (2022), while also extending the decompositions of triangulated planar graphs of Van den Heuvel, Ossona de Mendez, Quiroz, Rabinovich and Siebertz (2017), to planar graphs of arbitrary girth. Note that these decompositions are the precursors of the graph product structure theorem of planar graphs. We give improved bounds for χsub(Gp)\chi_{\textrm{sub}}(G^p) for all pp, whenever GG has bounded treewidth, bounded simple treewidth, bounded genus, or excludes a clique or biclique as a minor. For this we introduce a family of parameters which form a gradation between the strong and the weak colouring numbers. We give upper bounds for these parameters for graphs coming from such classes. Finally, we give a 2-approximation algorithm for the subchromatic number of graphs coming from any fixed class with bounded layered cliquewidth. In particular, this implies a 2-approximation algorithm for the subchromatic number of powers GpG^p of graphs coming from any fixed class with bounded layered treewidth (such as the class of planar graphs). This algorithm works even if the power pp and the graph GG is unknown.Comment: 21 pages, 2 figure

    On the Quantum Chromatic Numbers of Small Graphs

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    We make two contributions pertaining to the study of the quantum chromatic numbers of small graphs. Firstly, in an elegant paper, Man\v{c}inska and Roberson [\textit{Baltic Journal on Modern Computing}, 4(4), 846-859, 2016] gave an example of a graph G14G_{14} on 14 vertices with quantum chromatic number 4 and classical chromatic number 5, and conjectured that this is the smallest graph exhibiting a separation between the two parameters. We describe a computer-assisted proof of this conjecture, thereby resolving a longstanding open problem in quantum graph theory. Our second contribution pertains to the study of the rank-rr quantum chromatic numbers. While it can now be shown that for every rr, χq\chi_q and χq(r)\chi^{(r)}_q are distinct, few small examples of separations between these parameters are known. We give the smallest known example of such a separation in the form of a graph G21G_{21} on 21 vertices with χq(G21)=χq(2)(G21)=4\chi_q(G_{21}) = \chi^{(2)}_q(G_{21}) = 4 and ξ(G21)=χq(1)(G21)=χ(G21)=5 \xi(G_{21}) = \chi^{(1)}_q(G_{21}) = \chi(G_{21}) = 5. The previous record was held by a graph GmsgG_{msg} on 57 vertices that was first considered in the aforementioned paper of Man\v{c}inska and Roberson and which satisfies χq(Gmsg)=3\chi_q(G_{msg}) = 3 and χq(1)(Gmsg)=4\chi^{(1)}_q(G_{msg}) = 4. In addition, G21G_{21} provides the first provable separation between the parameters χq(1)\chi^{(1)}_q and χq(2)\chi^{(2)}_q. We believe that our techniques for constructing G21G_{21} and lower bounding its orthogonal rank could be of independent interest

    On graphs with no induced P5P_5 or K5eK_5-e

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    In this paper, we are interested in some problems related to chromatic number and clique number for the class of (P5,K5e)(P_5,K_5-e)-free graphs, and prove the following. (a)(a) If GG is a connected (P5,K5eP_5,K_5-e)-free graph with ω(G)7\omega(G)\geq 7, then either GG is the complement of a bipartite graph or GG has a clique cut-set. Moreover, there is a connected (P5,K5eP_5,K_5-e)-free imperfect graph HH with ω(H)=6\omega(H)=6 and has no clique cut-set. This strengthens a result of Malyshev and Lobanova [Disc. Appl. Math. 219 (2017) 158--166]. (b)(b) If GG is a (P5,K5eP_5,K_5-e)-free graph with ω(G)4\omega(G)\geq 4, then χ(G)max{7,ω(G)}\chi(G)\leq \max\{7, \omega(G)\}. Moreover, the bound is tight when ω(G){4,5,6}\omega(G)\notin \{4,5,6\}. This result together with known results partially answers a question of Ju and Huang [arXiv:2303.18003 [math.CO] 2023], and also improves a result of Xu [Manuscript 2022]. While the "Chromatic Number Problem" is known to be NPNP-hard for the class of P5P_5-free graphs, our results together with some known results imply that the "Chromatic Number Problem" can be solved in polynomial time for the class of (P5,K5eP_5,K_5-e)-free graphs which may be independent interest.Comment: This paper is dedicated to the memory of Professor Frederic Maffray on his death anniversar

    Near Optimal Colourability on Hereditary Graph Families

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    A graph family G\mathcal{G} is near optimal colourable if there is a constant number cc such that every graph GGG\in\mathcal{G} satisfies χ(G)max{c,ω(G)}\chi(G)\leq\max\{c,\omega(G)\}, where χ(G)\chi(G) and ω(G)\omega(G) are the chromatic number and clique number of GG, respectively. The near optimal colourable graph families together with the Lov{\'a}sz theta function are useful for the study of the chromatic number problems for hereditary graph families. In this paper, we investigate the near optimal colourability for (H1,H2H_1,H_2)-free graphs. Our main result is an almost complete characterization for the near optimal colourability for (H1,H2H_1,H_2)-free graphs with two exceptional cases, one of which is the celebrated Gy{\'a}rf{\'a}s conjecture. To obtain the result, we prove that the family of (2K2,P4Kn2K_2,P_4\vee K_n)-free graphs is near optimal colourable for every positive integer nn by inductive arguments.Comment: 11 pages, 1 figur

    Hardness Transitions of Star Colouring and Restricted Star Colouring

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    We study how the complexity of the graph colouring problems star colouring and restricted star colouring vary with the maximum degree of the graph. Restricted star colouring (in short, rs colouring) is a variant of star colouring. For kNk\in \mathbb{N}, a kk-colouring of a graph GG is a function f ⁣:V(G)Zkf\colon V(G)\to \mathbb{Z}_k such that f(u)f(v)f(u)\neq f(v) for every edge uvuv of GG. A kk-colouring of GG is called a kk-star colouring of GG if there is no path u,v,w,xu,v,w,x in GG with f(u)=f(w)f(u)=f(w) and f(v)=f(x)f(v)=f(x). A kk-colouring of GG is called a kk-rs colouring of GG if there is no path u,v,wu,v,w in GG with f(v)>f(u)=f(w)f(v)>f(u)=f(w). For kNk\in \mathbb{N}, the problem kk-STAR COLOURABILITY takes a graph GG as input and asks whether GG admits a kk-star colouring. The problem kk-RS COLOURABILITY is defined similarly. Recently, Brause et al. (Electron. J. Comb., 2022) investigated the complexity of 3-star colouring with respect to the graph diameter. We study the complexity of kk-star colouring and kk-rs colouring with respect to the maximum degree for all k3k\geq 3. For k3k\geq 3, let us denote the least integer dd such that kk-STAR COLOURABILITY (resp. kk-RS COLOURABILITY) is NP-complete for graphs of maximum degree dd by Ls(k)L_s^{(k)} (resp. Lrs(k)L_{rs}^{(k)}). We prove that for k=5k=5 and k7k\geq 7, kk-STAR COLOURABILITY is NP-complete for graphs of maximum degree k1k-1. We also show that 44-RS COLOURABILITY is NP-complete for planar 3-regular graphs of girth 5 and kk-RS COLOURABILITY is NP-complete for triangle-free graphs of maximum degree k1k-1 for k5k\geq 5. Using these results, we prove the following: (i) for k4k\geq 4 and dk1d\leq k-1, kk-STAR COLOURABILITY is NP-complete for dd-regular graphs if and only if dLs(k)d\geq L_s^{(k)}; and (ii) for k4k\geq 4, kk-RS COLOURABILITY is NP-complete for dd-regular graphs if and only if Lrs(k)dk1L_{rs}^{(k)}\leq d\leq k-1

    Coloring polygon visibility graphs and their generalizations

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    Curve pseudo-visibility graphs generalize polygon and pseudo- polygon visibility graphs and form a hereditary class of graphs. We prove that every curve pseudo-visibility graph with clique number ω has chromatic number at most 3 · 4ω−1. The proof is carried through in the setting of ordered graphs; we identify two conditions satisfied by every curve pseudo- visibility graph (considered as an ordered graph) and prove that they are sufficient for the claimed bound. The proof is algorithmic: both the clique number and a coloring with the claimed number of colors can be computed in polynomial time
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