arXiv subjects

L. Sunil Chandran

Publications and source records attributed to L. Sunil Chandran.

At least 19 recordsLinked to original sources

arXiv2026

Generalized Zykov's Theorem

For a simple graph $G$, let $n$ denote its number of vertices, and let $N(G,K_t)$ denote the number of copies of $K_t$ in $G$. Zykov's theorem (1949) asserts that for any $K_{r+1}$-free graph and $t \geq 2$, \[ N(G,K_t) \leq \binom{r}{t}\left(\frac{n}{r}\right)^t. \] We generalize Zykov's bound within a vertex-based localization framework. For each vertex $v \in V(G)$, let $c(v)$ denote the order of the largest clique containing $v$. Then \[ N(G,K_t) \leq n^{t-1} \sum_{v \in V(G)} \frac{1}{c(v)^t}\binom{c(v)}{t}. \] Moreover, when $G$ contains a copy of $K_t$, equality holds if and only if $G$ is a regular complete multipartite graph. Note that if we impose the condition that $G$ is $K_{r+1}$-free, then $c(v) \leq r$ for all $v \in V(G)$, and the monotonicity of $s \mapsto \binom{s}{t}/s^t$ gives Zykov's bound.

math.CO↗

arXiv2026

The boxicity of the compressed zero divisor graph of the ring of integers modulo N

The boxicity of a graph $G$, denoted by $box(G)$, is the minimum integer $d\geq 0$ such that $G$ is the intersection graph of axis-parallel boxes in $\mathbb{R}^d$. The class of zero divisor graphs introduced by Beck (1988) is a popular class of graphs and has been studied extensively by several researchers. Suppose $Z(R)$ is the set of zero divisors of a ring $R$. The zero divisor graph $Γ(R)$ for a ring $R $ is defined as the graph with the vertex set $V(Γ(R))=Z(R)$ and $E(Γ(R))=\{\{x,y\}\colon x,y\in Z(R)\text{ with }x\neq y\text{ and }x y=0\}$. One can define an equivalence relation $\sim$ on $V(Γ(R))$ such that for vertices $x$ and $y$, one has $x\sim y$ if and only if $x$ and $y$ have the same annihilator, i.e., $Ann(x)=Ann(y)$. The compressed zero divisor graph $Γ_E(R)$ for a ring $R$ is the simple graph obtained from $Γ(R)$ by retaining exactly one vertex from each equivalence class induced by $\sim$. In this paper, we completely answer two open questions posed in Discrete Applied Mathematics 391 (2026), pp. 127-136. Let $N=\prod_{i=1}^a p_i^{n_i}$ be the prime factorization of a positive integer $N$ and let $\mathbb{Z}_N$ be the ring of integers modulo $N$. We determine the exact boxicity of the compressed zero divisor graph $Γ_E(\mathbb{Z}_N)$. We show that when $a\geq 2$, $box(Γ_E(\mathbb{Z}_N))= a-1$ if and only if one of the following is true: $(i)$ $a\geq 2$ and $N$ is the product of two coprime integers $x$ and $y$ such that $x$ is a square-free integer and $y$ is the cube of a prime number; $(ii)$ $a\geq 3$ and $N$ is square-free; $(iii)$ $a\geq 2$, $N$ is cube-free, not square-free, and contains at least one prime divisor $p_i$ such that $n_i=1$. If $a=2$ and $n_1=n_2=1$, then $Γ_{E}(\mathbb{Z}_N)$ is a clique, and so, $box(Γ_{E}(\mathbb{Z}_N))=0$. In all other cases, $box(Γ_{E}(\mathbb{Z}_N))=a$.

cs.DM↗

arXiv2026

Boxicity and Cubicity of Divisor Graphs and Power Graphs

The \textit{boxicity} (\textit{cubicity}) of an undirected graph $Γ$ is the smallest non-negative integer $k$ such that $Γ$ can be represented as the intersection graph of axis-parallel rectangular boxes (unit cubes) in $\mathbb{R}^k$. An undirected graph is classified as a \textit{comparability graph} if it is isomorphic to the comparability graph of some partial order. This paper studies boxicity and cubicity for subclasses of comparability graphs. We initiate the study of boxicity and cubicity of a special class of algebraically defined comparability graphs, namely the \textit{power graphs}. The power graph of a group is an undirected graph whose vertex set is the group itself, with two elements being adjacent if one is a power of the other. We analyse the case when the underlying groups of power graphs are cyclic. Another important family of comparability graphs is \textit{divisor graphs}, which arises from a number-theoretically defined poset, namely the \textit{divisibility poset}. We consider a subclass of divisor graphs, denoted by $D(n)$, where the vertex set is the set of positive divisors of a natural number $n$. We first show that to study the boxicity (cubicity) of the power graph of the cyclic group of order $n$, it is sufficient to study the boxicity (cubicity) of $D(n)$. We derive estimates, tight up to a factor of $2$, for the boxicity and cubicity of $D(n)$. The exact estimates hold good for power graphs of cyclic groups.

math.CO↗

arXiv2026

Off-diagonal Rado numbers for $x+y+c=z$ and $x+y+k=z$

The study of Ramsey-type problems for linear equations originated with Schur's theorem and was later placed in a systematic framework by Richard Rado. In the off-diagonal setting, one fixes a pair of linear equations $(\mathcal{E}_1, \mathcal{E}_2)$ and seeks the least integer $N$ such that every red--blue coloring of $\{1,2,\dots,N\}$ contains either a red solution to $\mathcal{E}_1$ or a blue solution to $\mathcal{E}_2$. This threshold integer is referred to as the off-diagonal Rado number of the system $(\mathcal{E}_1, \mathcal{E}_2)$. In this work, we study the discrete and continuous two-color off-diagonal Rado numbers for the nonhomogeneous linear equations $x+y+c=z$ and $x+y+k=z$, where $c\leq k$. In the discrete setting, $c$ and $k$ are nonnegative integers, whereas in the continuous setting, they are nonnegative real numbers. We determine the exact discrete and continuous two-color off-diagonal Rado numbers for this pair of shifted Schur equations.

math.CO↗

arXiv2026

Graph Burning: Bounds and Hardness

Graph burning is a discrete-time process that models the propagation of information in a network. Given an undirected graph whose vertices are initially unburned, the process evolves in discrete rounds. At each round, an unburned vertex is selected and burned, while any unburned vertex adjacent to a vertex burned in the previous round also becomes burned. The burning number of a graph is the minimum number of steps to burn all its vertices. The BURNING NUMBER PROBLEM asks whether the burning number of an input graph $G$ is at most $k$. In this paper, we investigate the graph burning problem from both algorithmic and structural viewpoints. Although the problem is known to be NP-complete on interval graphs, we strengthen this result by proving that it remains NP-complete even when restricted to connected proper interval graphs. We also study the burning number of $P_k$-free graphs. Motivated by the well-known burning number conjecture, which states that every connected graph of order $n$ has burning number at most $\lceil \sqrt{n}~\rceil$, we establish an improved upper bound for connected $P_k$-free graphs and show that this bound is tight up to an additive constant of $1$. Finally, we study two variants of the problem: edge burning and total burning. We establish fundamental relationships between these variants and the classical burning, and we determine the computational complexity of the corresponding decision problems.

math.CO↗

arXiv2026

Hardness of Burning Number Problem on Regular Graphs

The Burning Number Problem (BNP) models the spread of information or contagion in a network through a discrete-time process on a graph. At each step, one new vertex is selected as a burning source, while fire simultaneously spreads from previously burned vertices to their neighbors. The burning number of a graph is the minimum number of steps required to burn all vertices. The decision version asks whether the burning number is at most a given integer $k$. BNP is known to be NP-complete even on restricted graph classes such as path forests. We study BNP on connected regular graphs, a natural and previously unexplored graph class. We prove that BNP is NP-complete on connected cubic graphs, and moreover APX-hard under this restriction. We further show that BNP remains APX-hard on connected $d$-regular graphs for every fixed $d \geq 4$.

cs.DS↗

arXiv2026

Parameterized algorithms for $k$-Inversion

Inversion of a directed graph $D$ with respect to a vertex subset $Y$ is the directed graph obtained from $D$ by reversing the direction of every arc whose endpoints both lie in $Y$. More generally, the inversion of $D$ with respect to a tuple $(Y_1, Y_2, \ldots, Y_\ell)$ of vertex subsets is defined as the directed graph obtained by successively applying inversions with respect to $Y_1, Y_2, \ldots, Y_\ell$. Such a tuple is called a \emph{decycling family} of $D$ if the resulting graph is acyclic. In the \textsc{$k$-Inversion} problem, the input consists of a directed graph $D$ and an integer $k$, and the task is to decide whether $D$ admits a decycling family of size at most $k$. Alon et al.\ (SIAM J.\ Discrete Math., 2024) proved that the problem is NP-complete for every fixed value of $k$, thereby ruling out XP algorithms, and presented a fixed-parameter tractable (FPT) algorithm parameterized by $k$ for tournament inputs. In this paper, we generalize their algorithm to a broader variant of the problem on tournaments and subsequently use this result to obtain an FPT algorithm for \textsc{$k$-Inversion} when the underlying undirected graph of the input is a block graph. Furthermore, we obtain an algorithm for \textsc{$k$-Inversion} on general directed graphs with running time $2^{O(\mathrm{tw}(k + \mathrm{tw}))} \cdot n^{O(1)}$, where $\mathrm{tw}$ denotes the treewidth of the underlying graph.

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arXiv2026

Localization: A Framework to Generalize Extremal Graph Problems

Extremal graph theory studies the maximum or minimum number of subgraphs isomorphic to a prescribed graph under given constraints. \textit{Localization} has recently emerged as a framework that refines such problems by assigning extremal quantities locally (to vertices or edges) and then aggregating them. This perspective not only recovers classical results but also leads to sharper bounds. A classical result states that a connected planar graph with a finite girth $g$ satisfies \begin{equation*} m \leq \frac{g}{g-2}(n-2) \end{equation*} Wood~\cite{wood} derived upper bounds on the number of $K_t$-cliques in graphs of bounded maximum degree, expressed in terms of both the number of vertices and the number of edges: \begin{align*} ex(n,K_t,K_{1,d+1}) \leq \frac{n}{d+1}\binom{d+1}{t} \\ mex(m,K_t,K_{1,d+1}) \leq \frac{m}{\binom{d+1}{2}}\binom{d+1}{t} \end{align*} More recently, Chakraborty and Chen~\cite{CHAKRABORTI2024103955} established a similar upper bound for graphs with bounded path length: \begin{equation*} mex(m,K_t,P_{r+1}) \leq \frac{m}{\binom{r}{2}}\binom{r}{t} \end{equation*} In this paper, we employ the localization framework to improve these bounds and provide structural characterizations of the extremal graphs attaining them.

math.CO↗

arXiv2026

Off-diagonal Rado number for $x+y+c=z$ and $x+qy=z$

Ramsey-type problems for linear equations began with Schur's theorem and were systematically generalized by Richard Rado. In the off-diagonal framework for two colors, one considers two different linear equations $(\mathcal{E}_1,\mathcal{E}_2)$ and determines the minimum integer $N$ for which any red-blue coloring of $\{1,2,...,N\}$ forces either a red solution of the equation $\mathcal{E}_1$ or a blue solution of the equation $\mathcal{E}_2$. In this work, we study off-diagonal Rado numbers for non-homogeneous linear equations of the forms $x+y+c=z$ and $x+qy=z$. We determine the exact two-color off-diagonal Rado number $R_2(c,q)$ associated with this system of equations.

math.CO↗

arXiv2025

Vertex-Based Localization of Generalized Turán Problems

Let $\mathcal{F}$ be a family of graphs. A graph is called $\mathcal{F}$-free if it does not contain any member of $\mathcal{F}$. Generalized Turán problems aim to maximize the number of copies of a graph $H$ in an $n$-vertex $\mathcal{F}$-free graph. This maximum is denoted by $ex(n, H, \mathcal{F})$. When $H \cong K_2$, it is simply denoted by $ex(n,F)$. Erdős and Gallai established the bounds $ex(n, P_{k+1}) \leq \frac{n(k-1)}{2}$ and $ex(n, C_{\geq k+1}) \leq \frac{k(n-1)}{2}$. This was later extended by Luo \cite{luo2018maximum}, who showed that $ex(n, K_s, P_{k+1}) \leq \frac{n}{k} \binom{k}{s}$ and $ex(n, K_s, C_{\geq k+1}) \leq \frac{n-1}{k-1} \binom{k}{s}$. Let $N(G,K_s)$ denote the number of copies of $K_s$ in $G$. In this paper, we use the vertex-based localization framework, introduced in \cite{adak2025vertex}, to generalize Luo's bounds. In a graph $G$, for each $v \in V(G)$, define $p(v)$ to be the length of the longest path that contains $v$. We show that \[N(G,K_s) \leq \sum_{v \in V(G)} \frac{1}{p(v)+1}{p(v)+1\choose s} = \frac{1}{s}\sum_{v \in V(G)}{p(v) \choose s-1}\] We strengthen the cycle bound from \cite{luo2018maximum} as follows: In graph $G$, for each $v \in V(G)$, let $c(v)$ be the length of the longest cycle that contains $v$, or $2$ if $v$ is not part of any cycle. We prove that \[N(G,K_s) \leq \left(\sum_{v\in V(G)}\frac{1}{c(v)-1}{c(v) \choose s}\right) - \frac{1}{c(u)-1}{c(u) \choose s}\] where $c(u)$ denotes the circumference of $G$. Furthermore, we characterize the class of extremal graphs that attain equality for these bounds. We provide full proofs for the cases $s = 1$ and $s \geq 3$, while the case $s = 2$ follows from the result in \cite{adak2025vertex}. We also conclude with a generalization of a result by Balister-Bollobás-Riordan-Schelp \cite{BALISTER2003366}.

math.CO↗

arXiv2025

CNFs and DNFs with Exactly $k$ Solutions

Model counting is a fundamental problem that consists of determining the number of satisfying assignments for a given Boolean formula. The weighted variant, which computes the weighted sum of satisfying assignments, has extensive applications in probabilistic reasoning, network reliability, statistical physics, and formal verification. A common approach for solving weighted model counting is to reduce it to unweighted model counting, which raises an important question: {\em What is the minimum number of terms (or clauses) required to construct a DNF (or CNF) formula with exactly $k$ satisfying assignments?} In this paper, we establish both upper and lower bounds on this question. We prove that for any natural number $k$, one can construct a monotone DNF formula with exactly $k$ satisfying assignments using at most $O(\sqrt{\log k}\log\log k)$ terms. This construction represents the first $o(\log k)$ upper bound for this problem. We complement this result by showing that there exist infinitely many values of $k$ for which any DNF or CNF representation requires at least $Ω(\log\log k)$ terms or clauses. These results have significant implications for the efficiency of model counting algorithms based on formula transformations.

cs.DM↗

arXiv2025

Boxicity of Zero Divisor Graphs

A $d$-dimensional box is the cartesian product $R_i\times\cdots\times R_d$ where each $R_i$ is a closed interval on the real line. The boxicity of a graph, denoted as $box(G)$, is the minimum integer $d\geq 0$ such that $G$ is the intersection graph of a collection of $d$-dimensional boxes. The study of graph classes associated with algebraic structures is a fascinating area where graph theory and algebra meet. A well-known class of graphs associated with rings is the class of zero divisor graphs introduced by Beck in 1988. Since then, this graph class has been studied extensively by several researchers. Denote by $Z(R)$ the set of zero divisors of a ring $R$. The zero divisor graph $Γ(R)$ for a ring $R$ is defined as the graph with the vertex set $V(Γ(R))=Z(R)$ and $E(Γ(R))=\{\{a_i,a_j\}:a_ia_j\in Z(R)\text{ and }a_ia_j=0 \}$. Let $N=Π_{i=1}^ap_i^{n_i}$ be the prime factorization of $N$. In Discrete Applied Mathematics 365 (2025), pp. 260-269, it was shown that $box(Γ(\mathbb{Z}_N))\leqΠ_{i=1}^a(n_i+1)-Π_{i=1}^a(\lfloor n_i/2\rfloor+1)-1$. In this paper we exactly determine the boxicity of $Γ(\mathbb{Z}_N)$: We show that when $N\equiv 2\pmod 4$ and $N$ is not divisible by $p^3$ for any prime divisor $p$, we have $box(Γ(\mathbb{Z}_N))=a-1$. Otherwise $box(Γ(\mathbb{Z}_N))=a$. Suppose $R$ is a non-zero commutative ring with identity that is also a reduced ring and let $k$ be the size of the set of minimal prime ideals of $R$. In the same paper, it was showed that $box(Γ(R))\leq 2^k-2$. We improve this result by showing $\lfloor k/2\rfloor\leq box(Γ(R))\leq k$ with the same assumption on $R$. In this paper we also show that $a-1\leq\dim_{TH}(Γ(\mathbb{Z}_N))\leq a$ and $\lfloor k/2\rfloor\leq\dim_{TH}(Γ(R))\leq k$, where $\dim_{TH}$ is another dimensional parameter associated with graphs known as the threshold dimension.

cs.DM↗

arXiv2025

Vertex-Based Localization of Turán's Theorem

Let $G$ be a simple graph with $n$ vertices and $m$ edges. According to Turán's theorem, if $G$ is $K_{r+1}$-free, then $m \leq |E(T(n, r))|,$ where $T(n, r)$ denotes the Turán graph on $n$ vertices with a maximum clique of order $r$. A limitation of this statement is that it does not give an expression in terms of $n$ and $r$. A widely used version of Turán's theorem states that for an $n$-vertex $K_{r+1}$-free graph, $m \leq \left\lfloor \frac{n^2(r-1)}{2r} \right\rfloor.$ Though this bound is often more convenient, it is not the same as the original statement. In particular, the class of extremal graphs for this bound, say $\mathcal{S}$, is a proper subset of the set of Turán graphs. In this paper, we generalize this result as follows: For each $v \in V(G)$, let $c(v)$ be the order of the largest clique that contains $v$. We show that \[ m \leq \left\lfloor\frac{n}{2}\sum_{v\in V(G)}\frac{c(v)-1}{c(v)}\right\rfloor\] Furthermore, we characterize the class of extremal graphs that attain equality in this bound. Interestingly, this class contains two extra non-Turán graphs other than the graphs in $\mathcal{S}$.

math.CO↗

arXiv2025

Vertex-Based Localization of Erdős-Gallai Theorems for Paths and Cycles

For a simple graph $G$, let $n$ and $m$ denote the number of vertices and edges in $G$, respectively. The Erdős-Gallai theorem for paths states that in a simple $P_k$-free graph, $m \leq \frac{n(k-1)}{2}$, where $P_k$ denotes a path with length $k$ (that is, with $k$ edges). In this paper, we generalize this result as follows: For each $v \in V(G)$, let $p(v)$ be the length of the longest path that contains $v$. We show that \[m \leq \sum_{v \in V(G)} \frac{p(v)}{2}\] The Erdős-Gallai theorem for cycles states that in a simple graph $G$ with circumference (that is, the length of the longest cycle) at most $k$, we have $m \leq \frac{k(n-1)}{2}$. We strengthen this result as follows: For each $v \in V(G)$, let $c(v)$ be the length of the longest cycle that contains $v$, or $2$ if $v$ is not part of any cycle. We prove that \[m \leq \left( \sum_{v \in V(G)} \frac{c(v)}{2} \right) - \frac{c(u)}{2}\] where $c(u)$ denotes the circumference of $G$. \newline Furthermore, we characterize the class of extremal graphs that attain equality in these bounds.

math.CO↗

arXiv2024

Total Domination, Separated Clusters, CD-Coloring: Algorithms and Hardness

Domination and coloring are two classic problems in graph theory. The major focus of this paper is the CD-COLORING problem which combines the flavours of domination and colouring. Let $G$ be an undirected graph. A proper vertex coloring of $G$ is a $cd-coloring$ if each color class has a dominating vertex in $G$. The minimum integer $k$ for which there exists a $cd-coloring$ of $G$ using $k$ colors is called the cd-chromatic number, $χ_{cd}(G)$. A set $S\subseteq V(G)$ is a total dominating set if any vertex in $G$ has a neighbor in $S$. The total domination number, $γ_t(G)$ of $G$ is the minimum integer $k$ such that $G$ has a total dominating set of size $k$. A set $S\subseteq V(G)$ is a $separated-cluster$ if no two vertices in $S$ lie at a distance 2 in $G$. The separated-cluster number, $ω_s(G)$, of $G$ is the maximum integer $k$ such that $G$ has a separated-cluster of size $k$. In this paper, first we explore the connection between CD-COLORING and TOTAL DOMINATION. We prove that CD-COLORING and TOTAL DOMINATION are NP-Complete on triangle-free $d$-regular graphs for each fixed integer $d\geq 3$. We also study the relationship between the parameters $χ_{cd}(G)$ and $ω_s(G)$. Analogous to the well-known notion of `perfectness', here we introduce the notion of `cd-perfectness'. We prove a sufficient condition for a graph $G$ to be cd-perfect (i.e. $χ_{cd}(H)= ω_s(H)$, for any induced subgraph $H$ of $G$) which is also necessary for certain graph classes (like triangle-free graphs). Here, we propose a generalized framework via which we obtain several exciting consequences in the algorithmic complexities of special graph classes. In addition, we settle an open problem by showing that the SEPARATED-CLUSTER is polynomially solvable for interval graphs.

cs.DS↗

arXiv2024

Two Results on LPT: A Near-Linear Time Algorithm and Parcel Delivery using Drones

The focus of this paper is to increase our understanding of the Longest Processing Time First (LPT) heuristic. LPT is a classical heuristic for the fundamental problem of uniform machine scheduling. For different machine speeds, LPT was first considered by Gonzalez et al (SIAM J. Computing, 1977). Since then, extensive work has been done to improve the approximation factor of the LPT heuristic. However, all known implementations of the LPT heuristic take $O(mn)$ time, where $m$ is the number of machines and $n$ is the number of jobs. In this work, we come up with the first near-linear time implementation for LPT. Specifically, the running time is $O((n+m)(\log^2{m}+\log{n}))$. Somewhat surprisingly, the result is obtained by mapping the problem to dynamic maintenance of lower envelope of lines, which has been well studied in the computational geometry community. Our second contribution is to analyze the performance of LPT for the Drones Warehouse Problem (DWP), which is a natural generalization of the uniform machine scheduling problem motivated by drone-based parcel delivery from a warehouse. In this problem, a warehouse has multiple drones and wants to deliver parcels to several customers. Each drone picks a parcel from the warehouse, delivers it, and returns to the warehouse (where it can also get charged). The speeds and battery lives of the drones could be different, and due to the limited battery life, each drone has a bounded range in which it can deliver parcels. The goal is to assign parcels to the drones so that the time taken to deliver all the parcels is minimized. We prove that the natural approach of solving this problem via the LPT heuristic has an approximation factor of $ϕ$, where $ϕ\approx 1.62$ is the golden ratio.

cs.DS↗

arXiv2024

Graph-theoretic insights on the constructability of complex entangled states

The most efficient automated way to construct a large class of quantum photonic experiments is via abstract representation of graphs with certain properties. While new directions were explored using Artificial intelligence and SAT solvers to find such graphs, it becomes computationally infeasible to do so as the size of the graph increases. So, we take an analytical approach and introduce the technique of local sparsification on experiment graphs, using which we answer a crucial open question in experimental quantum optics, namely whether certain complex entangled quantum states can be constructed. This provides us with more insights into quantum resource theory, the limitation of specific quantum photonic systems and initiates the use of graph-theoretic techniques for designing quantum physics experiments.

quant-ph↗

arXiv2024

Krenn-Gu conjecture for sparse graphs

Greenberger-Horne-Zeilinger (GHZ) states are quantum states involving at least three entangled particles. They are of fundamental interest in quantum information theory, and the construction of such states of high dimension has various applications in quantum communication and cryptography. They are of fundamental interest in quantum information theory, and the construction of such states of high dimension has various applications in quantum communication and cryptography. Krenn, Gu and Zeilinger discovered a correspondence between a large class of quantum optical experiments which produce GHZ states and edge-weighted edge-coloured multi-graphs with some special properties called the \emph{GHZ graphs}. On such GHZ graphs, a graph parameter called \emph{dimension} can be defined, which is the same as the dimension of the GHZ state produced by the corresponding experiment. Krenn and Gu conjectured that the dimension of any GHZ graph with more than $4$ vertices is at most $2$. An affirmative resolution of the Krenn-Gu conjecture has implications for quantum resource theory. On the other hand, the construction of a GHZ graph on a large number of vertices with a high dimension would lead to breakthrough results. In this paper, we study the existence of GHZ graphs from the perspective of the Krenn-Gu conjecture and show that the conjecture is true for graphs of vertex connectivity at most 2 and for cubic graphs. We also show that the minimal counterexample to the conjecture should be $4$-connected. Such information could be of great help in the search for GHZ graphs using existing tools like PyTheus. While the impact of the work is in quantum physics, the techniques in this paper are purely combinatorial, and no background in quantum physics is required to understand them.

quant-ph↗