Turán Density of $2$-Edge-Colored Bipartite Graphs with Application on $\{2, 3\}$-Hypergraphs

We consider the Turán problems of $2$-edge-colored graphs. A $2$-edge-colored graph $H=(V, E_r, E_b)$ is a triple consisting of the vertex set $V$, the set of red edges $E_r$ and the set of blue edges $E_b$ where $E_r$ and $E_b$ do not have to be disjoint. The Turán density $\pi(H)$ of $H$ is defined to be $\lim_{n\to\infty} \max_{G_n}h_n(G_n)$, where $G_n$ is chosen among all possible $2$-edge-colored graphs on $n$ vertices containing no $H$ as a sub-graph and $h_n(G_n)=(|E_r(G)|+|E_b(G)|)/\binom{n}{2}$ is the formula to measure the edge density of $G_n$. We will determine the Turán densities of all $2$-edge-colored bipartite graphs. We also give an important application on the Turán problems of $\{2, 3\}$-hypergraphs.

A steady-state grouping genetic algorithm for the rainbow spanning tree problem

Given a connected, undirected and edge-colored graph, the rainbow spanning tree (RSF) problem aims to find a rainbow spanning forest with the minimum number of rainbow trees, where a rainbow tree is a connected acyclic subgraph of the graph whose each edge is associated with a different color. This problem is $NP$-Hard and finds several applications in distinguishing among various types of connections. Being a grouping problem, this paper proposes a steady-state grouping genetic algorithm (SSGGA) for the RSF problem. To the best of our knowledge, this is the first work on steady-state grouping genetic algorithm for this problem. While keeping in view of grouping aspects of the problem, each individual, in the proposed SSGGA, is encoded as a group of rainbow trees, and accordingly a problem-specific crossover operator is designed. Moreover, SSGGA uses the idea of two steps in its replacement scheme. All such elements of SSGGA coordinate effectively and overall help in finding high quality solutions. Computational results obtained over a set of benchmark instances show that overall SSGGA, in terms of solution quality, is superior to all other existing approaches in the literature for this problem.

On d-distance equitable chromatic number of some graphs

An assignment of distinct colors [Formula: see text] to the vertices [Formula: see text] and [Formula: see text] of a graph [Formula: see text] such that the distance between [Formula: see text] and [Formula: see text] is at most [Formula: see text] is called [Formula: see text]-distance coloring of [Formula: see text]. Suppose [Formula: see text] are the color classes of [Formula: see text]-distance coloring and [Formula: see text] for any [Formula: see text], then [Formula: see text] is [Formula: see text]-distance equitable colored graph. In this paper, we obtain [Formula: see text]-distance chromatic number and [Formula: see text]-distance equitable chromatic number of graphs like [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text].

Tight Upper Bound of the 3-Total-Rainbow Index for 2-(Edge-)Connected Graphs

Let [Formula: see text] be an edge-colored graph with order [Formula: see text] and [Formula: see text] be a fixed integer satisfying [Formula: see text]. For a vertex set [Formula: see text] of at least two vertices, a tree containing the vertices of [Formula: see text] in [Formula: see text] is called an [Formula: see text]-tree. The [Formula: see text]-tree [Formula: see text] is a total-rainbow [Formula: see text]-tree if the elements of [Formula: see text], except for the vertex set [Formula: see text], have distinct colors. A total-colored graph [Formula: see text] is said to be total-rainbow [Formula: see text]-tree connected if for every set [Formula: see text] of [Formula: see text] vertices in [Formula: see text], there exists a total-rainbow [Formula: see text]-tree in [Formula: see text], while the total-coloring of [Formula: see text] is called a [Formula: see text]-total-rainbow coloring. The [Formula: see text]-total-rainbow index of a nontrivial connected graph [Formula: see text], denoted by [Formula: see text], is the smallest number of colors needed in a [Formula: see text]-total-rainbow coloring of [Formula: see text]. In this paper, we show a sharp upper bound for [Formula: see text], where [Formula: see text] is a 2-connected or 2-edge-connected graph.

Chromatic spectrum of some classes of 2-regular bipartite colored graphs

Chromatic spectrum of a colored graph G is a multiset of eigenvalues of colored adjacency matrix of G. The nullity of a disconnected graph is equal to sum of nullities of its components but we show that this result does not hold for colored graphs. In this paper, we investigate the chromatic spectrum of three different classes of 2-regular bipartite colored graphs. In these classes of graphs, it is proved that the nullity of G is not sum of nullities of components of G. We also highlight some important properties and conjectures to extend this problem to general graphs.

Total Proper Connection and Graph Operations

A graph is said to be total-colored if all the edges and vertices of the graph are colored. A path in a total-colored graph is a total proper path if (i) any two adjacent edges on the path differ in color, (ii) any two internal adjacent vertices on the path differ in color, and (iii) any internal vertex of the path differs in color from its incident edges on the path. A total-colored graph is called total-proper connected if any two vertices of the graph are connected by a total proper path of the graph. For a connected graph [Formula: see text], the total proper connection number of [Formula: see text], denoted by [Formula: see text], is defined as the smallest number of colors required to make [Formula: see text] total-proper connected. In this paper, we study the total proper connection number for the graph operations. We find that 3 is the total proper connection number for the join, the lexicographic product and the strong product of nearly all graphs. Besides, we study three kinds of graphs with one factor to be traceable for the Cartesian product as well as the permutation graphs of the star and traceable graphs. The values of the total proper connection number for these graphs are all [Formula: see text].

Genericity of chaos for colored graphs

To each colored graph one can associate its closure in the universal space of isomorphism classes of pointed colored graphs, and this subspace can be regarded as a generalized subshift. Based on this correspondence, we introduce two definitions for chaotic (colored) graphs, one of them analogous to Devaney’s. We show the equivalence of our two novel definitions of chaos, proving their topological genericity in various subsets of the universal space.

A dynamic programming algorithm for solving the k-Color Shortest Path Problem

Several variants of the classical Constrained Shortest Path Problem have been presented in the literature so far. One of the most recent is the k-Color Shortest Path Problem ($$k$$ k -CSPP), that arises in the field of transmission networks design. The problem is formulated on a weighted edge-colored graph and the use of the colors as edge labels allows to take into account the matter of path reliability while optimizing its cost. In this work, we propose a dynamic programming algorithm and compare its performances with two solution approaches: a Branch and Bound technique proposed by the authors in their previous paper and the solution of the mathematical model obtained with CPLEX solver. The results gathered in the numerical validation evidenced how the dynamic programming algorithm vastly outperformed previous approaches.

Some algorithmic results for finding compatible spanning circuits in edge-colored graphs

A compatible spanning circuit in a (not necessarily properly) edge-colored graph G is a closed trail containing all vertices of G in which any two consecutively traversed edges have distinct colors. Sufficient conditions for the existence of extremal compatible spanning circuits (i.e., compatible Hamilton cycles and Euler tours), and polynomial-time algorithms for finding compatible Euler tours have been considered in previous literature. More recently, sufficient conditions for the existence of more general compatible spanning circuits in specific edge-colored graphs have been established. In this paper, we consider the existence of (more general) compatible spanning circuits from an algorithmic perspective. We first show that determining whether an edge-colored connected graph contains a compatible spanning circuit is an NP-complete problem. Next, we describe two polynomial-time algorithms for finding compatible spanning circuits in edge-colored complete graphs. These results in some sense give partial support to a conjecture on the existence of compatible Hamilton cycles in edge-colored complete graphs due to Bollobás and Erdős from the 1970s.