Monochromatic Triangles Research Articles

Monochromatic triangles in the max-norm plane

For all non-degenerate triangles T, we determine the minimum number of colors needed to color the plane such that no max-norm isometric copy of T is monochromatic.

A threading path to a Ramsey number

The article describes a collaboration between an established artist and a professional mathematician. This at the same time accidental and, in the spirit of Ramsey theory, unavoidable collaboration has resulted in creating an artwork inspired by an edge 3-colouring of the complete graph on sixteen vertices that avoids monochromatic triangles.

Variations on Ramsey numbers and minimum numbers of monochromatic triangles in line $2$-colorings of configurations

Variations on Ramsey numbers and minimum numbers of monochromatic triangles in line $2$-colorings of configurations

A Note on the Geometry of Closed Loops

In this paper, we utilize the Ramsey theory to investigate the geometrical characteristics of closed contours. We begin by examining a set of six points arranged on a closed contour and connected as a complete graph. We assign the downward-pointing edges a red color, while coloring the remaining edges green. Our analysis establishes that the curve must contain at least one monochromatic triangle. This finding has practical applications in the study of dynamical billiards. Our second result is derived from the Jordan curve theorem and the Ramsey theorem. Finally, we discuss Ramsey constructions arising from differential geometry. Applications of the Ramsey theory are discussed.

The pigeonhole principle and multicolor Ramsey numbers

A standard proof of Schur's Theorem yields that any $r$-coloring of $\{1,2,\dots,R_r-1\}$ yields a monochromatic solution to $x+y=z$, where $R_r$ is the classical $r$-color Ramsey number, the minimum $N$ such that any $r$-coloring of a complete graph on $N$ vertices yields a monochromatic triangle. We explore generalizations and modifications of this result in higher dimensional integer lattices, showing in particular that if $k\geq d+1$, then any $r$-coloring of $\{1,2,\dots,R_r(k)^d-1\}^d$ yields a monochromatic solution to $x_1+\cdots+x_{k-1}=x_k$ with $\{x_1,\dots,x_d\}$ linearly independent, where $R_r(k)$ is the analogous Ramsey number in which triangles are replaced by complete graphs on $k$ vertices. We also obtain computational results and examples in the case $d=2$, $k=3$, and $r\in\{2,3,4\}$.

Identification Conditions for the Solvability of NP-Complete Problems for the Class of Prefractal Graphs

Modern network systems (unmanned aerial vehicles groups, social networks, network production chains, transport and logistics networks, communication networks, and cryptocurrency networks) are distinguished by their multielement nature and the dynamics of connections between their elements. A number of discrete problems on the construction of optimal substructures of network systems described in the form of various classes of graphs are NP-complete problems. In this case, the variability and dynamism of the structures of network systems lead to an additional complication of the search for solutions to discrete optimization problems. At the same time, for some subclasses of dynamic graphs that are used to model the structures of network system, conditions for the solvability of a number of NP-complete problems can be distinguished. This subclass of dynamic graphs includes prefractal graphs. This article investigates NP-complete problems on prefractal graphs: a Hamiltonian cycle, a skeleton with the maximum number of pendant vertices, a monochromatic triangle, a clique, and an independent set. The conditions under which it is possible to obtain an answer about the existence of and construct polynomial (when fixing the number of seed vertices) algorithms for finding solutions for some problems are identified.

Color neighborhood union conditions for proper edge-pancyclicity of edge-colored complete graphs

Let G be an edge-colored complete graph on n vertices such that there exist at least n distinct colors on edges incident to every pair of its vertices. In this paper, we first show that every edge of G with n≥6k−19 is contained in a properly colored cycle of length k. Further, we prove that if G contains no monochromatic triangles, then there exists a properly colored path of length l for every 1≤l≤n−1 between each pair of vertices of G and every vertex of G is contained in a properly colored cycle of length k for any 3≤k≤n.

Bounding Monochromatic Triangles Using Squares

Summary Goodman gave the lower bound on the number of monochromatic triangles of the complete graph on n vertices when the edges are colored with two colors. We reprove his result by giving an algebraic reformulation and utilizing that squares of real expressions are nonnegative.

QUBO formulations of three NP problems

With the recent advancements in the field of combinatorial optimization, QUBO (quadratic unconstrained binary optimization) has become a topic of intense interest, as a problem format acceptable to several hardware heuristics such as quantum and digital annealers. Accordingly, transforming hard mathematical problems into QUBO problems is of particular importance. In this paper, we restate three problems in NP-complete and NP-hard classes as QUBO problems with proofs of equivalence. Our problems of interest are the numerical three-dimensional matching problem, the monochromatic triangle problem and the optimum facility location problem. Furthermore, we comment on the feasibility of these formulations on sophisticated QUBO solving platforms.

Properly colored cycles in edge-colored complete graphs without monochromatic triangle: A vertex-pancyclic analogous result

A properly colored cycle (path) in an edge-colored graph is a cycle (path) with consecutive edges assigned distinct colors. A monochromatic triangle is a cycle of length 3 with the edges assigned a same color. It is known that every edge-colored complete graph without monochromatic triangle always contains a properly colored Hamilton path. In this paper, we investigate the existence of properly colored cycles in edge-colored complete graphs when monochromatic triangles are forbidden. We obtain a vertex-pancyclic analogous result combined with a characterization of all the exceptions. As a consequence, we partially confirm a structural conjecture given by Bollobás and Erdős (1976) and an algorithmic conjecture given by Gutin and Kim (2009).

Extremal problems and results related to Gallai-colorings

A Gallai-coloring (Gallai-k-coloring) is an edge-coloring (with colors from {1,2,…,k}) of a complete graph without rainbow triangles. Given a graph H and a positive integer k, the k-colored Gallai-Ramsey number GRk(H) is the minimum integer n such that every Gallai-k-coloring of the complete graph Kn contains a monochromatic copy of H. In this paper, we consider two extremal problems related to Gallai-k-colorings. First, we determine upper and lower bounds for the maximum number of edges that are not contained in any rainbow triangle or monochromatic triangle in a k-edge-coloring of Kn. Second, for n≥GRk(K3), we determine upper and lower bounds for the minimum number of monochromatic triangles in a Gallai-k-coloring of Kn, yielding the exact value for k=3. Furthermore, we determine the Gallai-Ramsey number GRk(K4+e) for the graph on five vertices consisting of a K4 with a pendant edge.

Identification Conditions for the Solvability of NP-complete Problems for the Class of Pre-fractal Graphs

Modern network systems (unmanned aerial vehicles groups, social networks, network production chains, transport and logistics networks, communication networks, cryptocurrency networks) are distinguished by their multi-element nature and the dynamics of connections between its elements. A number of discrete problems on the construction of optimal substructures of network systems described in the form of various classes of graphs are NP-complete problems. In this case, the variability and dynamism of the structures of network systems leads to an "additional" complication of the search for solutions to discrete optimization problems. At the same time, for some subclasses of dynamical graphs, which are used to model the structures of network systems, conditions for the solvability of a number of NP-complete problems can be distinguished. This subclass of dynamic graphs includes pre-fractal graphs. The article investigates NP-complete problems on pre-fractal graphs: a Hamiltonian cycle, a skeleton with the maximum number of pendant vertices, a monochromatic triangle, a clique, an independent set. The conditions under which for some problems it is possible to obtain an answer about the existence and to construct polynomial (when fixing the number of seed vertices) algorithms for finding solutions are identified.

Normal approximation and fourth moment theorems for monochromatic triangles

AbstractGiven a graph sequence denote by T3(Gn) the number of monochromatic triangles in a uniformly random coloring of the vertices of Gn with colors. In this paper we prove a central limit theorem (CLT) for T3(Gn) with explicit error rates, using a quantitative version of the martingale CLT. We then relate this error term to the well‐known fourth‐moment phenomenon, which, interestingly, holds only when the number of colors satisfies . We also show that the convergence of the fourth moment is necessary to obtain a Gaussian limit for any , which, together with the above result, implies that the fourth‐moment condition characterizes the limiting normal distribution of T3(Gn), whenever . Finally, to illustrate the promise of our approach, we include an alternative proof of the CLT for the number of monochromatic edges, which provides quantitative rates for the results obtained in [7].

Proper vertex-pancyclicity of edge-colored complete graphs without joint monochromatic triangles

In an edge-colored graph (G,c), let dc(v) denote the number of colors on the edges incident with a vertex v of G and δc(G) denote the minimum value of dc(v) over all vertices v∈V(G). A cycle of (G,c) is called proper if any two adjacent edges of the cycle have distinct colors. An edge-colored graph (G,c) on n≥3 vertices is called properly vertex-pancyclic if each vertex of (G,c) is contained in a proper cycle of length ℓ for every ℓ with 3≤ℓ≤n. Fujita and Magnant conjectured that every edge-colored complete graph on n≥3 vertices with δc(G)≥n+12 is properly vertex-pancyclic. Chen, Huang and Yuan partially solve this conjecture by adding an extra condition that (G,c) does not contain any monochromatic triangle. In this paper, we show that this conjecture is true if the edge-colored complete graph contain no joint monochromatic triangles.

Extremal triangle-free and odd-cycle-free colourings of uncountable graphs

The optimality of the Erdős–Rado theorem for pairs is witnessed by the colouring $$\Delta_\kappa : [2^\kappa]^2 \rightarrow \kappa$$ recording the least point of disagreement between two functions. This colouring has no monochromatic triangles or, more generally, odd cycles. We investigate a number of questions investigating the extent to which $$\Delta_\kappa$$ is an extremal such triangle-free or odd-cycle-free colouring. We begin by introducing the notion of $$\Delta$$ -regressive and almost $$\Delta$$ -regressive colourings and studying the structures that must appear as monochromatic subgraphs for such colourings. We also consider the question as to whether $$\Delta_\kappa$$ has the minimal cardinality of any maximal triangle-free or odd-cycle-free colouring into $$\kappa$$ . We resolve the question positively for odd-cycle-free colourings.

On Almost Empty Monochromatic Triangles and Convex Quadrilaterals in Colored Point Sets

Let S be a set of N points on the plane in general position, colored arbitrarily with c colors ($$c\in {\mathbb {N}}$$). A subset of points is said to be monochromatic if all its points have the same color. In this paper we show that if N is sufficiently large, then S contains a monochromatic triangle with vertices in S and containing at most $$c-3$$ points in its interior $$(c \ge 4)$$. The particular case $$c=4$$ solves the conjecture of Basu, Bhattacharya, and Das in [6]. For the case of two colors, it is still unknown whether every sufficiently large bichromatic point set contains an empty monochromatic convex quadrilateral [11]. In this direction we show that if S is sufficiently large, then it always contains a convex monochromatic quadrilateral with at most $$2c-3$$ points in its interior $$(c \ge 2)$$. We also show a general result on the number of empty convex quadrilaterals with disjoint interiors. Let P be a set of n points in general position. It is straightforward to see that if the elements of P are the vertices of a convex polygon, $$n \ge 4$$, P always has $$\left\lceil \frac{n-3}{2} \right\rceil $$ empty convex quadrilaterals with disjoint interiors. In this paper we prove that any point set P in general position has at least $$\left\lfloor \frac{n-3}{2}\right\rfloor $$ interior disjoint empty convex quadrilaterals. We also give direct consequences of this theorem related to simultaneously flippable edges in triangulations and convex decompositions of point sets. We show that for any point set P there is always a triangulation that has $$\left\lfloor \frac{n-3}{2}\right\rfloor $$ simultaneously flippable edges. We also show that there is always a convex decomposition of P consisting of triangles and convex quadrilaterals with at most $${3n-2h\over 2}$$ elements, where h is the number of points in the convex hull of P.

White-Box vs. Black-Box Complexity of Search Problems

Ramsey theory assures us that in any graph there is a clique or independent set of a certain size, roughly logarithmic in the graph size. But how difficult is it to find the clique or independent set? If the graph is given explicitly, then it is possible to do so while examining a linear number of edges. If the graph is given by a black-box, where to figure out whether a certain edge exists the box should be queried, then a large number of queries must be issued. But what if one is given a program or circuit for computing the existence of an edge? This problem was raised by Buss and Goldberg and Papadimitriou in the context of TFNP, search problems with a guaranteed solution. We examine the relationship between black-box complexity and white-box complexity for search problems with guaranteed solution such as the above Ramsey problem. We show that under the assumption that collision-resistant hash function exists (which follows from the hardness of problems such as factoring, discrete-log, and learning with errors) the white-box Ramsey problem is hard and this is true even if one is looking for a much smaller clique or independent set than the theorem guarantees. This is also true for the colorful Ramsey problem where one is looking, say, for a monochromatic triangle. In general, one cannot hope to translate all black-box hardness for TFNP into white-box hardness: we show this by adapting results concerning the random oracle methodology and the impossibility of instantiating it. Another model we consider is that of succinct black-box, where the complexity of an algorithm is measured as a function of the description size of the object in the box (and no limitation on the computation time). In this case, we show that for all TFNP problems there is an efficient algorithm with complexity proportional to the description size of the object in the box times the solution size. However, for promise problems this is not the case. Finally, we consider the complexity of graph property testing in the white-box model. We show a property that is hard to test even when one is given the program for computing the graph (under the appropriate assumptions such as hardness of Decisional Diffie-Hellman). The hard property is whether the graph is a two-source extractor.

Colouring set families without monochromatic k-chains

A coloured version of classic extremal problems dates back to Erdős and Rothschild, who in 1974 asked which n-vertex graph has the maximum number of 2-edge-colourings without monochromatic triangles. They conjectured that the answer is simply given by the largest triangle-free graph. Since then, this new class of coloured extremal problems has been extensively studied by various researchers. In this paper we pursue the Erdős–Rothschild versions of Sperner's Theorem, the classic result in extremal set theory on the size of the largest antichain in the Boolean lattice, and Erdős' extension to k-chain-free families.Given a family F of subsets of [n], we define an (r,k)-colouring of F to be an r-colouring of the sets without any monochromatic k-chains F1⊂F2⊂…⊂Fk. We prove that for n sufficiently large in terms of k, the largest k-chain-free families also maximise the number of (2,k)-colourings. We also show that the middle level, ([n]⌊n/2⌋), maximises the number of (3,2)-colourings, and give asymptotic results on the maximum possible number of (r,k)-colourings whenever r(k−1) is divisible by three.

Proper vertex-pancyclicity of edge-colored complete graphs without monochromatic triangles

In an edge-colored graph (G,c), let dc(v) be the number of colors on the edges incident to vertex v and let δc(G) be the minimum value of dc(v) over all vertices v∈V(G). A cycle of (G,c) is called proper if any two adjacent edges of the cycle have distinct colors. An edge-colored graph (G,c) on n≥3 vertices is called properly vertex-pancyclic if each vertex of (G,c) is contained in a proper cycle of length l for every l with 3≤l≤n. Fujita and Magnant conjectured that every edge-colored complete graph on n≥3 vertices with δc(G)≥n+12 is properly vertex-pancyclic. We show that this conjecture is true if the edge-colored complete graph has no monochromatic triangles.

EXPANDER GRAPHS AND SIEVING IN COMBINATORIAL STRUCTURES

We prove a general large-sieve statement in the context of random walks on subgraphs of a given graph. This can be seen as a generalization of previously known results where one performs a random walk on a group enjoying a strong spectral gap property. In such a context the point is to exhibit a strong uniform expansion property for a suitable family of Cayley graphs on quotients. In our combinatorial approach, this is replaced by a result of Alon–Roichman about expanding properties of random Cayley graphs. Applying the general setting we show, for instance, that with high probability (in a strong explicit sense) random coloured subsets of integers contain monochromatic (nonempty) subsets summing to $0$, and that a random colouring of the edges of a complete graph contains a monochromatic triangle.