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We consider a variant of the densest subgraph problem in networks with single or multiple edge attributes. For example, in a social network, the edge attributes may describe the type of relationship between users, such as friends, family,…
A vertex of degree one in a tree is called an end vertex and a vertex of degree at least three is called a branch vertex. For a graph $G$, let $\sigma_2$ be the minimum degree sum of two nonadjacent vertices in $G$. We consider tree…
An exact $(k,d)$-coloring of a graph $G$ is a coloring of its vertices with $k$ colors such that each vertex $v$ is adjacent to exactly $d$ vertices having the same color as $v$. The exact $d$-defective chromatic number, denoted…
We consider the minimum spanning tree problem with predictions, using the weight-arrival model, i.e., the graph is given, together with predictions for the weights of all edges. Then the actual weights arrive one at a time and an…
For an edge-colored graph $G$, the minimum color degree of $G$ means the minimum number of colors on edges which are adjacent to each vertex of $G$. We prove that if $G$ is an edge-colored graph with minimum color degree at least $5$ then…
The Balanced Connected Subgraph problem (BCS) was recently introduced by Bhore et al. (CALDAM 2019). In this problem, we are given a graph $G$ whose vertices are colored by red or blue. The goal is to find a maximum connected subgraph of…
A {\it rainbow matching} in an edge-colored graph is a matching in which all the edges have distinct colors. Wang asked if there is a function f(\delta) such that a properly edge-colored graph G with minimum degree \delta and order at least…
Colouring sparse graphs under various restrictions is a theoretical problem of significant practical relevance. Here we consider the problem of maximizing the number of different colours available at the nodes and their neighbourhoods,…
Let $G$ be a finite or infinite graph and $m(G)$ the minimum number of vertices moved by the non-identity automorphisms of $G$. We are interested in bounds on the supremum $\Delta(G)$ of the degrees of the vertices of $G$ that assure the…
An edge-colored graph $G$ is {\em rainbow connected} if any two vertices are connected by a path whose edges have distinct colors. The {\em rainbow connection} of a connected graph $G$, denoted $rc(G)$, is the smallest number of colors that…
Graph coloring, also known as vertex coloring, considers the problem of assigning colors to the nodes of a graph such that adjacent nodes do not share the same color. The optimization version of the problem concerns the minimization of the…
Graph coloring is a computationally difficult problem, and currently the best known classical algorithm for $k$-coloring of graphs on $n$ vertices has runtimes $\Omega(2^n)$ for $k\ge 5$. The list coloring problem asks the following more…
In the Exact Matching Problem (EM), we are given a graph equipped with a fixed coloring of its edges with two colors (red and blue), as well as a positive integer $k$. The task is then to decide whether the given graph contains a perfect…
The graph coloring problem asks for an assignment of the minimum number of distinct colors to vertices in an undirected graph with the constraint that no pair of adjacent vertices share the same color. The problem is a thoroughly studied…
Given a directed graph $G=(V,A)$, the Directed Maximum Leaf Spanning Tree problem asks to compute a directed spanning tree (i.e., an out-branching) with as many leaves as possible. By designing a Branch-and-Reduced algorithm combined with…
This is the second paper in a series of two. The goal of the series is to give a polynomial time algorithm for the $4$-coloring problem and the $4$-precoloring extension problem restricted to the class of graphs with no induced six-vertex…
Let $c:E(G)\to [k]$ be an edge-coloring of a graph $G$, not necessarily proper. For each vertex $v$, let $\bar{c}(v)=(a_1,\ldots,a_k)$, where $a_i$ is the number of edges incident to $v$ with color $i$. Reorder $\bar{c}(v)$ for every $v$ in…
All Colors Shortest Path problem defined on an undirected graph aims at finding a shortest, possibly non-simple, path where every color occurs at least once, assuming that each vertex in the graph is associated with a color known in…
A hamiltonian coloring $c$ of a graph $G$ of order $n$ is a mapping $c$ : $V(G) \rightarrow \{0,1,2,...\}$ such that $D(u, v)$ + $|c(u) - c(v)|$ $\geq$ $n-1$, for every two distinct vertices $u$ and $v$ of $G$, where $D(u, v)$ denotes the…
A new algorithm for exactly sampling from the set of proper colorings of a graph is presented. This is the first such algorithm that has an expected running time that is guaranteed to be linear in the size of a graph with maximum degree \(…