Related papers: Finding and counting vertex-colored subtrees
In vertex recoloring, we are given $n$ vertices with their initial coloring, and edges arrive in an online fashion. The algorithm must maintain a valid coloring by recoloring vertices, at a cost. The problem abstracts a scenario of job…
A properly edge-colored graph is a graph with a coloring of its edges such that no vertex is incident to two or more edges of the same color. A subgraph is called rainbow if all its edges have different colors. The problem of finding…
A graph is said to be {\it total-colored} if all the edges and the vertices of the graph are colored. A path in a total-colored graph is a {\it total monochromatic path} if all the edges and internal vertices on the path have the same…
The Colouring problem asks whether the vertices of a graph can be coloured with at most $k$ colours for a given integer $k$ in such a way that no two adjacent vertices receive the same colour. A graph is $(H_1,H_2)$-free if it has no…
We consider vertex colorings of graphs in which adjacent vertices have distinct colors. A graph is $s$-chromatic if it is colorable in $s$ colors and any coloring of it uses at least $s$ colors. The forcing chromatic number $F(G)$ of an…
A vertex-colored graph $G$ is {\it rainbow vertex-connected} if any pair of vertices in $G$ are connected by a path whose internal vertices have distinct colors, which was introduced by Krivelevich and Yuster. The {\it rainbow…
In a connected simple graph G = (V,E), each vertex of V is colored by a color from the set of colors C={c1, c2,..., c_{\alpha}}$. We take a subset S of V, such that for every vertex v in V\S, at least one vertex of the same color is present…
An (h,s,t)-representation of a graph G consists of a collection of subtrees of a tree T, where each subtree corresponds to a vertex of G such that (i) the maximum degree of T is at most h, (ii) every subtree has maximum degree at mots s,…
An edge-colored graph $G$ is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that…
We study weighted edge coloring of graphs, where we are given an undirected edge-weighted general multi-graph $G := (V, E)$ with weights $w : E \rightarrow [0, 1]$. The goal is to find a proper weighted coloring of the edges with as few…
In a colouring of a graph, a vertex is b-chromatic if it is adjacent to a vertex of every other colour. We consider four well-studied colouring problems: b-Chromatic Number, Tight b-Chromatic Number, Fall Chromatic Number and Fall…
Let $G=(V,E)$ be a graph on $n$ vertices and $R$ be a set of pairs of vertices in $V$ called \emph{requests}. A \emph{multicut} is a subset $F$ of $E$ such that every request $xy$ of $R$ is cut by $F$, \i.e. every $xy$-path of $G$…
Structural graph parameters, such as treewidth, pathwidth, and clique-width, are a central topic of study in parameterized complexity. A main aim of research in this area is to understand the "price of generality" of these widths: as we…
For a given number of colors, $s$, the guessing number of a graph is the (base $s$) logarithm of the cardinality of the largest family of colorings of the vertex set of the graph such that the color of each vertex can be determined from the…
Consider a graph $G = (V,E)$ and a coloring $c$ of vertices with colors from $[\ell]$. A vertex $v$ is said to be happy with respect to $c$ if $c(v) = c(u)$ for all neighbors $u$ of $v$. Further, an edge $(u,v)$ is happy if $c(u) = c(v)$.…
Subgraph detection has recently been one of the most studied problems in the CONGEST model of distributed computing. In this work, we study the distributed complexity of problems closely related to subgraph detection, mainly focusing on…
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…
In this paper we investigate the colorful components framework, motivated by applications emerging from comparative genomics. The general goal is to remove a collection of edges from an undirected vertex-colored graph $G$ such that in the…
We introduce a variant of the graph coloring problem, which we denote as {\sc Budgeted Coloring Problem} (\bcp). Given a graph $G$, an integer $c$ and an ordered list of integers $\{b_1, b_2, \ldots, b_c\}$, \bcp asks whether there exists a…
Coloring is a notoriously hard problem, and even more so in the online setting, where each arriving vertex has to be colored immediately and irrevocably. Already on trees, which are trivially two-colorable, it is impossible to achieve…