Edge-colored Graph Research Articles

Polytopality of maniplexes

Given an abstract polytope P, its flag graph is the edge-coloured graph whose vertices are the flags of P and whose i-edges correspond to i-adjacent flags. Flag graphs of polytopes are maniplexes. On the other hand, a maniplex need not be the flag graph of a polytope. It is natural to ask when does a maniplex is the flag graph of a polytope. In this paper we give necessary and sufficient conditions (in terms of graphs) on a maniplex to be (isomorphic to) the flag graph of a polytope. For this, given a maniplex M, we define a poset PM and determine when is PM an abstract polytope. Moreover, in such case, we show that M is isomorphic to the flag graph of PM.

Designing deterministic polynomial-space algorithms by color-coding multivariate polynomials

We introduce an enhancement of color coding to design deterministic polynomial-space parameterized algorithms. Our approach aims at reducing the number of random choices by exploiting the special structure of a solution. Using our approach, we derive polynomial-space O⁎(3.86k)-time (exponential-space O⁎(3.41k)-time) deterministic algorithm for k-Internal Out-Branching, improving upon the previously fastest exponential-space O⁎(5.14k)-time algorithm for this problem. (The notation O⁎ hides polynomial factors.) We also design polynomial-space O⁎((2e)k+o(k))-time (exponential-space O⁎(4.32k)-time) deterministic algorithms for k-ColorfulOut-Branching on arc-colored digraphs and k-Colorful Perfect Matching on planar edge-colored graphs. In k-Colorful Out-Branching, given an arc-colored digraph D, decide whether D has an out-branching with arcs of at least k colors. k-Colorful Perfect Matching is defined similarly. To obtain our polynomial-space algorithms, we show that (n,k,αk)-splitters (α⩾1) and in particular (n,k)-perfect hash families can be enumerated one by one with polynomial delay using polynomial space.

Optimization of scheduling system for plant watering using electric cars in agro techno park

Agro Techno Park in University of Jember is a special area used for the development of agriculture, livestock and fishery. In this plantation, the process of watering the plants is according to the frequency of each plant needs. This research develops the optimization of plant watering scheduling system using edge coloring of graph. This research was conducted in 3 stages, namely, data collection phase, analysis phase, and system development stage. The collected data was analyzed and then converted into a graph by using bipartite adjacency matrix representation. The development phase is conducted to build a web-based watering schedule optimization system. The result of this research showed that the schedule system is optimal because it can maximize the use of all electric cars to water the plants and minimize the number of idle cars.

Large rainbow matchings in semi-strong edge-colorings of graphs

The lower bounds for the size of maximum rainbow matching in properly edge-colored graphs have been studied deeply during the last decades. An edge-coloring of a graph [Formula: see text] is called a strong edge-coloring if each path of length at most three is rainbow. Clearly, the strong edge-coloring is a natural generalization of the proper one. Recently, Babu et al. considered the problem in the strongly edge-colored graphs. In this paper, we introduce a semi-strong edge-coloring of graphs and consider the existence of large rainbow matchings in it.

Minimum rainbow [formula omitted]-decompositions of graphs

Given graphs G and H, we consider the problem of decomposing a properly edge-colored graph G into few parts consisting of rainbow copies of H and single edges. We establish a close relation to the previously studied problem of minimum H-decompositions, where an edge coloring does not matter and one is merely interested in decomposing graphs into copies of H and single edges.

English

An edge-colored graph G is proper connected if every pair of vertices is connected by a proper path. The proper connection number of a connected graph G, denoted by pc(G), is the smallest number of colors that are needed to color the edges of G in order to make it proper connected. In this paper, we obtain the sharp upper bound for pc(G) of a general bipartite graph G and a series of extremal graphs. Additionally, we give a proper 2-coloring for a connected bipartite graph G having δ(G) ≥ 2 and a dominating cycle or a dominating complete bipartite subgraph, which implies pc(G) = 2. Furthermore, we get that the proper connection number of connected bipartite graphs with δ ≥ 2 and diam(G) ≤ 4 is two.

Budgeted Colored Matching Problems

The budgeted colored matching problem (BCM) is a weighted matching problem on edge-colored graphs with additional edge costs. Budget constraints require the accumulated cost of same-colored edges not to exceed a corresponding specified budget. We show the strong NP-hardness of the BCM on bipartite graphs with uniform edge weights, costs and budgets using a reduction from (3,B2)-SAT. Subsequently, we propose a dynamic program for series-parallel graphs with pseudo-polynomial run time for a fixed number of budget constraints. As an extension we show how this algorithm can be used to solve the BCM on trees using a graph transformation.

Topology in colored tensor models via crystallization theory

The aim of this paper is twofold. On the one hand, it provides a review of the links between random tensor models, seen as quantum gravity theories, and the PL-manifolds representation by means of edge-colored graphs (crystallization theory). On the other hand, the core of the paper is to establish results about the topological and geometrical properties of the Gurau-degree (or G-degree) of the represented manifolds, in relation with the motivations coming from physics. In fact, the G-degree appears naturally in higher dimensional tensor models as the quantity driving their 1/N expansion, exactly as it happens for the genus of surfaces in the two-dimensional matrix model setting. In particular, the G-degree of PL-manifolds is proved to be finite-to-one in any dimension, while in dimension 3 and 4 a series of classification theorems are obtained for PL-manifolds represented by graphs with a fixed G-degree. All these properties have specific relevance in the tensor models framework, showing a direct fruitful interaction between tensor models and discrete geometry, via crystallization theory.

Rainbow Saturation and Graph Capacities

The $t$-colored rainbow saturation number ${rsat}_t(n,F)$ is the minimum size of a $t$-edge-colored graph on $n$ vertices that contains no rainbow copy of $F$, but the addition of any missing edge ...

Monochromatic homeomorphically irreducible trees in $2$-edge-colored complete graphs

Monochromatic homeomorphically irreducible trees in $2$-edge-colored complete graphs

On Ryser’s Conjecture for $t$-Intersecting and Degree-Bounded Hypergraphs

A famous conjecture (usually called Ryser's conjecture) that appeared in the PhD thesis of his student, J. R. Henderson, states that for an $r$-uniform $r$-partite hypergraph $\mathcal{H}$, the inequality $\tau(\mathcal{H})\le(r-1)\!\cdot\! \nu(\mathcal{H})$ always holds. This conjecture is widely open, except in the case of $r=2$, when it is equivalent to Kőnig's theorem, and in the case of $r=3$, which was proved by Aharoni in 2001.Here we study some special cases of Ryser's conjecture. First of all, the most studied special case is when $\mathcal{H}$ is intersecting. Even for this special case, not too much is known: this conjecture is proved only for $r\le 5$ by Gyárfás and Tuza. For $r>5$ it is also widely open.Generalizing the conjecture for intersecting hypergraphs, we conjecture the following. If an $r$-uniform $r$-partite hypergraph $\mathcal{H}$ is $t$-intersecting (i.e., every two hyperedges meet in at least $t<r$ vertices), then $\tau(\mathcal{H})\le r-t$. We prove this conjecture for the case $t> r/4$.Gyárfás showed that Ryser's conjecture for intersecting hypergraphs is equivalent to saying that the vertices of an $r$-edge-colored complete graph can be covered by $r-1$ monochromatic components.Motivated by this formulation, we examine what fraction of the vertices can be covered by $r-1$ monochromatic components of different colors in an $r$-edge-colored complete graph. We prove a sharp bound for this problem.Finally we prove Ryser's conjecture for the very special case when the maximum degree of the hypergraph is two.

Finding disjoint paths on edge-colored graphs: more tractability results

The problem of finding the maximum number of vertex-disjoint uni-color paths in an edge-colored graph (called MaxCDP) has been recently introduced in literature, motivated by applications in social network analysis. In this paper we investigate how the complexity of the problem depends on graph parameters (namely the number of vertices to remove to make the graph a collection of disjoint paths and the size of the vertex cover of the graph), which makes sense since graphs in social networks are not random and have structure. The problem was known to be hard to approximate in polynomial time and not fixed-parameter tractable (FPT) for the natural parameter. Here, we show that it is still hard to approximate, even in FPT-time. Finally, we introduce a new variant of the problem, called MaxCDDP, whose goal is to find the maximum number of vertex-disjoint and color-disjoint uni-color paths. We extend some of the results of MaxCDP to this new variant, and we prove that unlike MaxCDP, MaxCDDP is already hard on graphs at distance two from disjoint paths.

Maximum Cuts in Edge-colored Graphs

The input of the Maximum Colored Cut problem consists of a graph G=(V,E) with an edge-coloring c:E→{1,2,3,…,p} and a positive integer k>0, and the question is whether G has a nontrivial edge cut using at least k colors. The Colorful Cut problem has the same input but asks for a nontrivial edge cut using all colors. Unlike what happens for the classical Maximum Cut problem, we prove that both problems are NP-complete even on complete, planar, or bounded treewidth graphs. Furthermore, we prove that Colorful Cut is NP-complete even when each color class induces a clique of size at most 3, but is trivially solvable when each color induces a K2. On the positive side, we prove that Maximum Colored Cut is fixed-parameter tractable when parameterized by either k or p, and that it admits a cubic kernel in both cases.

Large Monochromatic Components in Edge Colored Graphs with a Minimum Degree Condition

It is well-known that in every $k$-coloring of the edges of the complete graph $K_n$ there is a monochromatic connected component of order at least ${n\over k-1}$. In this paper we study an extension of this problem by replacing complete graphs by graphs of large minimum degree. For $k=2$ the authors proved that $\delta(G)\ge{3n\over 4}$ ensures a monochromatic connected component with at least $\delta(G)+1$ vertices in every $2$-coloring of the edges of a graph $G$ with $n$ vertices. This result is sharp, thus for $k=2$ we really need a complete graph to guarantee that one of the colors has a monochromatic connected spanning subgraph. Our main result here is that for larger values of $k$ the situation is different, graphs of minimum degree $(1-\epsilon_k)n$ can replace complete graphs and still there is a monochromatic connected component of order at least ${n\over k-1}$, in fact $$\delta(G)\ge \left(1 - \frac{1}{1000(k-1)^9}\right)n$$ suffices.Our second result is an improvement of this bound for $k=3$. If the edges of $G$ with $\delta(G)\geq {9n\over 10}$ are $3$-colored, then there is a monochromatic component of order at least ${n\over 2}$. We conjecture that this can be improved to ${7n\over 9}$ and for general $k$ we conjecture the following: if $k\geq 3$ and $G$ is a graph of order $n$ such that $\delta(G)\geq \left( 1 - \frac{k-1}{k^2}\right)n$, then in any $k$-coloring of the edges of $G$ there is a monochromatic connected component of order at least ${n\over k-1}$.

The (3, l)-Rainbow Edge-Index of Cartesian Product Graphs

For a graph G and a vertex subset [Formula: see text] of at least two vertices, an S-tree is a subgraph T of G that is a tree with [Formula: see text]. Two S-trees are said to be edge-disjoint if they have no common edge. Let [Formula: see text] denote the maximum number of edge-disjoint S-trees in G. For an integer K with [Formula: see text], the generalized k-edge-connectivity is defined as [Formula: see text]. An S-tree in an edge-colored graph is rainbow if no two edges of it are assigned the same color. Let [Formula: see text] and l be integers with [Formula: see text], the [Formula: see text]-rainbow edge-index [Formula: see text] of G is the smallest number of colors needed in an edge-coloring of G such that for every set S of k vertices of G, there exist l edge-disjoint rainbow S-trees.In this paper, we study the [Formula: see text]-rainbow edge-index of Cartesian product graphs and get a sharp upper bound for [Formula: see text] , where G and H are connected graphs with orders at least three, and [Formula: see text] denotes the Cartesian product of G and H.

Decomposing edge-colored graphs under color degree constraints

For an edge-colored graph G, the minimum color degree of G means the minimum number of colors on edges which are incident to each vertex of G. We prove that if G is an edge-colored graph with minimum color degree at least 5 then V (G) can be partitioned into two parts such that each part induces a subgraph with minimum color degree at least 2. We show this theorem by proving a much stronger form. Moreover, we point out an important relationship between our theorem and Bermond-Thomassen's conjecture in digraphs.

Minimum rainbow H-decompositions of graphs

Given graphs G and H, we consider the problem of decomposing a properly edge-colored graph G into few parts consisting of rainbow copies of H and single edges. We establish a close relation to the previously studied problem of minimum H-decompositions, where an edge coloring does not matter and one is merely interested in decomposing graphs into copies of H and single edges.

Rainbow connection number and graph operations

A path in an edge-colored graph G is rainbow if no two edges of the path are colored the same. An edge-colored graph G is rainbow connected if every two distinct vertices are connected by a rainbow path. The rainbow connection numberrc(G) of G is the smallest number of colors that are needed in order to make G rainbow connected. In this paper, we study bounds of rainbow connection number of some graph operations, such as the union of two graphs, adding edges, deleting edges, and adding vertices and edges. Moreover, we also study the following extremal problem. Let k and n be two integers such that 1≤k≤ℓ<n. Find the smallest integer f(n,k,ℓ) such that for each graph G of order n and diameter k, there exists an edge set F⊆E(G¯) satisfying |F|≤f(n,k,ℓ) and rc(G+F)≤ℓ.

Edge coloring of graphs embedded in a surface of nonnegative characteristic

Let G be a graph embeddable in a surface of nonnegative characteristic with maximum degree six. In this paper, we prove that if G contains no a vertex v which is contained in all cycles of lengths from 3 to 6, then G is of Class 1.

Parameterized complexity of the MINCCA problem on graphs of bounded decomposability

In an edge-colored graph, the cost incurred at a vertex on a path when two incident edges with different colors are traversed is called reload or changeover cost. The Minimum Changeover Cost Arborescence (MinCCA) problem consists in finding an arborescence with a given root vertex such that the total changeover cost of the internal vertices is minimized. It has been recently proved by Gözüpek et al. (2016) that the MinCCA problem when parameterized by the treewidth and the maximum degree of the input graph is in FPT. In this article we present the following hardness results for MinCCA:•the problem is W[1]-hard when parameterized by the vertex cover number of the input graph, even on graphs of degeneracy at most 3. In particular, it is W[1]-hard parameterized by the treewidth of the input graph, which answers the main open problem in the work of Gözüpek et al. (2016);•it is W[1]-hard on multigraphs parameterized by the tree-cutwidth of the input multigraph; and•it remains NP-hard on planar graphs even when restricted to instances with at most 6 colors and 0/1 symmetric costs, or when restricted to instances with at most 8 colors, maximum degree bounded by 4, and 0/1 symmetric costs.