Monochromatic Edges Research Articles

Defective coloring of hypergraphs

AbstractWe prove that the vertices of every ‐uniform hypergraph with maximum degree may be colored with colors such that each vertex is in at most monochromatic edges. This result, which is best possible up to the value of the constant , generalizes the classical result of Erdős and Lovász who proved the case.

Majority choosability of countable graphs

In any vertex coloring of a graph some edges have differently colored ends (bichromatic edges) and some are monochromatic. In a proper coloring all edges are bichromatic. In a majority coloring it is enough that for every vertex v, the number of monochromatic edges incident to v does not exceed the number of bichromatic edges incident to v. A well known result proved by Lovász asserts that every finite graph has a majority 2-coloring. A similar statement for countably infinite graphs is a challenging open problem, known as the Unfriendly Partition Conjecture.We consider a natural list variant of majority coloring. A graph is majorityk-choosable if it has a majority coloring from any lists of size k assigned arbitrarily to the vertices. We prove that every countable graph is majority 4-choosable. We also consider a natural analog of majority coloring for directed graphs. We prove that every countable digraph is also majority 4-choosable. We pose list and directed analogs of the Unfriendly Partition Conjecture, stating that every countable graph is majority 2-choosable and every countable digraph is majority 3-choosable.

Expected polynomial-time randomized algorithm for graph coloring problem

Given a graph G=(V,E) with n=|V| vertices, the graph coloring problem is defined as Find a color vectorC=(c(v)), where c(v)∈{1,2,…,n} denotes the color of vertex v∈V, such that no monochromatic edge exists in G and the span, the total number of distinct colors in C, is minimized. Since the problem is NP-complete, a greedy coloring is commonly used for solving it. Greedy coloring visits the vertices of G following an order S, and while visiting a vertex v, it puts the minimum color absent in all neighbors of v. We show that the orders producing span ≤k, where k≤n is a positive integer, can be partitioned into disjoint subsets of equivalent orders. Next, we propose a selective search (SS) algorithm, which takes ρ as an input parameter, selects ≥ρ orders each from a different set of equivalent orders with high probability, applies greedy coloring on them, and returns the color vector with minimum span. We analytically show that SS performs better than greedy coloring with high probability by evaluating the same number of orders. We propose an incremental search heuristic (ISH), which ρ1 times execute SS with parameter ρ2 and returns the color vector with minimum span. A parallel version of ISH called PISH is also proposed, executing ρ1 SS calls in parallel. We show that ISH hits an optimum coloring for a graph G=(V,E) with χ(G)(Δ(G)+1)≤ne in expected O(|V|+|E|) time and space complexities. We have evaluated ISH and PISH on 136 challenging benchmarks and shown that they significantly outperform 10 existing state-of-the-art algorithms. Finally, we validated our theoretical findings by evaluating ISH and greedy coloring on random graphs.

SAT-boosted Tabu Search for Coloring Massive Graphs

Graph coloring is the problem of coloring the vertices of a graph with as few colors as possible, avoiding monochromatic edges. It is one of the most fundamental NP-hard computational problems. For decades researchers have developed exact and heuristic methods for graph coloring. While methods based on propositional satisfiability (SAT) feature prominently among these exact methods, the encoding size is prohibitive for large graphs. For such graphs, heuristic methods have been proposed, with tabu search among the most successful ones. In this article, we enhance tabu search for graph coloring within the SAT-based local improvement (SLIM) framework. Our hybrid algorithm incrementally improves a candidate solution by repeatedly selecting small subgraphs and coloring them optimally with a SAT solver. This approach scales to dense graphs with several hundred thousand vertices and over 1.5 billion edges. Our experimental evaluation shows that our hybrid algorithm beats state-of-the-art methods on large dense graphs.

On small non-uniform hypergraphs without property B

For a given hypergraph H = (V, E) consider the sum q(H) of 2^{−|e|} over e ∈ E. Consider the class of hypergraphs whose smallest edge is of size n and for which every 2-colouring has a monochromatic edge. Let q(n) be the smallest value of q(H) in this class. We provide a survey of the known bounds on q(n) and make some minor refinements.

A 2-Bisection with Small Number of Monochromatic Edges of a Claw-Free Cubic Graph

A bisection of a graph G is a partition of its vertex set into two parts of the same cardinality. A k-bisection of G is a bisection of G such that every component of each part has at most k vertices. Cui and Liu proved that every claw-free cubic graph contains a 2-bisection. In this paper, we improve this result showing that every claw-free cubic graph contains a 2-bisection with bounded number of monochromatic edges, where a monochromatic edge of a 2-bisection is an edge connecting two vertices of the same part of the 2-bisection. We also prove that our bound is best possible for all claw-free cubic simple graphs.

The Complexity of Finding Fair Independent Sets in Cycles

Let G be a cycle graph and let \(V_1,\ldots,V_m\) be a partition of its vertex set into m sets. An independent set S of G is said to fairly represent the partition if \(|S \cap V_i| \geq \frac{1}{2} \cdot |V_i| -1\) for all \(i \in [m]\). It is known that for every cycle and every partition of its vertex set, there exists an independent set that fairly represents the partition (Aharoni et al. 2017). We prove that the problem of finding such an independent set is PPA-complete. As an application, we show that the problem of finding a monochromatic edge in a Schrijver graph, given a succinct representation of a coloring that uses fewer colors than its chromatic number, is PPA-complete as well. The work is motivated by the computational aspects of the `cycle plus triangles' problem and of its extensions.

Enumeration of rooted 3-connected bipartite planar maps

Abstract We provide the first solution to the problem of counting rooted 3-connected bipartite planar maps. Our starting point is the enumeration of bicoloured planar maps according to the number of edges and monochromatic edges, following Bernardi and Bousquet-Mélou [J. Comb. Theory Ser. B (2011)]. The decomposition of a map into 2- and 3-connected components allows us to obtain the generating functions of 2- and 3-connected bicoloured maps. Setting to zero the variable marking monochromatic edges we obtain the generating function of 3-connected bipartite maps, which is algebraic of degree 26. We deduce from it an asymptotic estimate for the number of 3-connected bipartite planar maps of the form t · n −5/2 γ n , where γ = ρ −1 ≈ 2.40958 and ρ ≈ 0.41501 is an algebraic number of degree 10.

Equitable Colorings of Hypergraphs with r Colors

The paper deals with a problem concerning equitable vertex colorings of uniform hypergraphs, i.e., colorings under which there are no monochromatic edges and simultaneously all the color classes have almost the same cardinalities. We obtain a new bound on the edge number of an n-uniform hypergraph that guarantees the existence of an equitable vertex coloring with r colors for this hypergraph.

A Ramsey variant of the Brown–Erdős–Sós conjecture

An $r$-uniform hypergraph ($r$-graph for short) is called linear if every pair of vertices belong to at most one edge. A linear $r$-graph is complete if every pair of vertices are in exactly one edge. The famous Brown-Erd\H{o}s-S\'os conjecture states that for every fixed $k$ and $r$, every linear $r$-graph with $\Omega(n^2)$ edges contains $k$ edges spanned by at most $(r-2)k+3$ vertices. As an intermediate step towards this conjecture, Conlon and Nenadov recently suggested to prove its natural Ramsey relaxation. Namely, that for every fixed $k$, $r$ and $c$, in every $c$-colouring of a complete linear $r$-graph, one can find $k$ monochromatic edges spanned by at most $(r-2)k+3$ vertices. We prove that this Ramsey version of the conjecture holds under the additional assumption that $r \geq r_0(c)$, and we show that for $c=2$ it holds for all $r\geq 4$.

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].

Monochromatic disconnection of graphs

For an edge-colored graph G, we call an edge–cut M of G monochromatic if the edges of M are colored with the same color. The graph G is called monochromatically disconnected if any two distinct vertices of G are separated by a monochromatic edge–cut. For a connected graph G, the monochromatic disconnection number of G, denoted by md(G), is the maximum number of colors that are needed in order to make G monochromatically disconnected. We show that almost all graphs have monochromatic disconnection numbers equal to 1. We also obtain the Nordhaus–Gaddum-type results for md(G).

On Rainbow-Cycle-Forbidding Edge Colorings of Finite Graphs

It is shown that whenever the edges of a connected simple graph on n vertices are colored with $$n-1$$ colors appearing so that no cycle in G is rainbow, there must be a monochromatic edge cut in G. From this it follows that such colorings of G can be represented, or ‘encoded,’ by full binary trees with n leaves, with vertices labeled by subsets of V(G), such that the leaf labels are singletons, the label of each non-leaf is the union of the labels of its children, and each label set induces a connected subgraph of G. It is also shown that $$n-1$$ is the largest integer for which the main theorem holds, for each n, although for some graphs a certain strengthening of the hypothesis makes the theorem conclusion true with $$n-1$$ replaced by $$n-2$$.

On Equitable Colorings of Hypergraphs

A two-coloring is said to be equitable if, on the one hand, there are no monochromatic edges (the coloring is regular) and, on the other hand, the cardinalities of color classes differ from one another by at most 1. It is proved that, for the existence of an equitable two-coloring, it suffices that the number of edges satisfy an estimate of the same order as that for a regular coloring. This result strengthens the previously known Radhakrishnan-Srinivasan theorem.

Equitable colorings of hypergraphs with few edges

The paper deals with an extremal problem concerning equitable colorings of uniform hypergraph. Recall that a vertex coloring of a hypergraph H is called proper if there are no monochromatic edges under this coloring. A hypergraph is said to be equitably r-colorable if there is a proper coloring with r colors such that the sizes of any two color classes differ by at most one. In the present paper we prove that if the number of edges |E(H)|≤0.01nlnnr−1rrn−1 then the hypergraph H is equitably r-colorable provided r<lnn5.

Equitable coloring of hypergraphs

A hypergraph is equitably k-colorable if its vertices can be partitioned into k sets/color classes in such a way that monochromatic edges are avoided and the number of vertices in any two color classes differs by at most one. We prove that the problem of equitable 2-coloring of hypergraphs is NP-complete even for 3-uniform hyperstars. Finally, we apply the method of dynamic programming for designing a polynomial-time algorithm to equitably k-color linear hypertrees, where k≥2 is fixed.

Tight bounds on the coefficients of partition functions via stability

Partition functions arise in statistical physics and probability theory as the normalizing constant of Gibbs measures and in combinatorics and graph theory as graph polynomials. For instance the partition functions of the hard-core model and monomer-dimer model are the independence and matching polynomials respectively.We show how stability results follow naturally from the recently developed occupancy method for maximizing and minimizing physical observables over classes of regular graphs, and then show these stability results can be used to obtain tight extremal bounds on the individual coefficients of the corresponding partition functions.As applications, we prove new bounds on the number of independent sets and matchings of a given size in regular graphs. For large enough graphs and almost all sizes, the bounds are tight and confirm the Upper Matching Conjecture of Friedland, Krop, and Markström and a conjecture of Kahn on independent sets for a wide range of parameters. Additionally we prove tight bounds on the number of q-colorings of cubic graphs with a given number of monochromatic edges, and tight bounds on the number of independent sets of a given size in cubic graphs of girth at least 5.

Extremes of the internal energy of the Potts model on cubic graphs

AbstractWe prove tight upper and lower bounds on the internal energy per particle (expected number of monochromatic edges per vertex) in the anti‐ferromagnetic Potts model on cubic graphs at every temperature and for all . This immediately implies corresponding tight bounds on the anti‐ferromagnetic Potts partition function. Taking the zero‐temperature limit gives new results in extremal combinatorics: the number of q‐colorings of a 3‐regular graph, for any , is maximized by a union of 's. This proves the d = 3 case of a conjecture of Galvin and Tetali.

Tight bounds on the coefficients of partition functions via stability

We show how to use the recently-developed occupancy method and stability results that follow easily from the method to obtain extremal bounds on the individual coefficients of the partition functions over various classes of bounded degree graphs.As applications, we prove new bounds on the number of independent sets and matchings of a given size in regular graphs. For large enough graphs and almost all sizes, the bounds are tight and confirm the Upper Matching Conjecture of Friedland, Krop, and Markström, and a conjecture of Kahn on independent sets for a wide range of parameters. Additionally we prove tight bounds on the number of q-colorings of cubic graphs with a given number of monochromatic edges, and tight bounds on the number of independent sets of a given size in cubic graphs of girth at least 5.

On Color Critical Graphs with Large Adaptable Chromatic Numbers

When the vertices and edges are colored with k colors, an edge is called monochromatic if the edge and the two vertices incident with it all have the same color. The adapted chromatic number of a graph G, $$\chi _{a}\left( G\right) ,$$ is the least integer k such that for each k-edge coloring of G the vertices of G can be colored with the same set of colors without creating any monochromatic edges. It is easy to see that $$\chi _{a}\left( G\right) \le \chi \left( G\right) $$ . While this bound is tight, all the known graphs attaining this bound are not color critical. It is known that if G is a critical graph, then $$\chi _{a}\left( G\right) \le \chi \left( G\right) -1$$ . In this article we construct a family of k-critical graphs whose adapted chromatic number is exactly one less than their chromatic number. This answers a question in Molloy and Thron (J Graph Theory 71:331–351, 2012). We also study the properties of graphs that are critical with respect to adaptable chromatic number.