Related papers: Make a graph singly connected by edge orientations
Coloring is one of the most famous problems in graph theory. The coloring problem on undirected graphs has been well studied, whereas there are very few results for coloring problems on directed graphs. An oriented k-coloring of an oriented…
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 $rc(G)$, is the smallest number of colors that are…
Let $G$ be an edge-colored connected graph. A path $P$ in $G$ is called a distance $\ell$-proper path if no two edges of the same color appear with fewer than $\ell$ edges in between on $P$. The graph $G$ is called $(k,\ell)$-proper…
A graph is said to be {\it total-colored} if all the edges and vertices of the graph are colored. A path in a total-colored graph is a {\it total proper path} if $(i)$ any two adjacent edges on the path differ in color, $(ii)$ any two…
An edge-colored graph $G$ is said to be rainbow connected if between each pair of vertices there exists a path which uses each color at most once. The rainbow connection number, denoted by $rc(G)$, is the minimum number of colors needed to…
We consider the worst-case query complexity of some variants of certain \cl{PPAD}-complete search problems. Suppose we are given a graph $G$ and a vertex $s \in V(G)$. We denote the directed graph obtained from $G$ by directing all edges in…
In the last years, connection concepts such as rainbow connection and proper connection appeared in graph theory and obtained a lot of attention. In this paper, we investigate the loose edge-connection of graphs. A connected edge-coloured…
Given a graph $G$, we say that an orientation $D$ of $G$ is a KT orientation if, for all $u, v \in V(D)$, there is at most one directed path (in any direction) between $u$ and $v$. Graphs that admit such orientations have been used by…
Ordered matchings, defined as graphs with linearly ordered vertices, where each vertex is connected to exactly one edge, play a crucial role in the area of ordered graphs and their homomorphisms. Therefore, we consider related problems from…
A connected ordering $(v_1, v_2, \ldots, v_n)$ of $V(G)$ is an ordering of the vertices such that $v_i$ has at least one neighbour in $\{v_1, \ldots, v_{i - 1}\}$ for every $i \in \{2, \ldots, n\}$. A connected greedy coloring (CGC for…
A path $P$ in an edge-colored graph $G$ is called \emph{a proper path} if no two adjacent edges of $P$ are colored the same, and $G$ is \emph{proper connected} if every two vertices of $G$ are connected by a proper path in $G$. The…
Let $G = (V, E)$ be a connected graph, and let $T$ in $V$ be a subset of vertices. An orientation of $G$ is called $T$-odd if any vertex $v \in V$ has odd in-degree if and only if it is in $T$. Finding a T -odd orientation of G can be…
Given a connected graph $G$, the metric (resp. edge metric) dimension of $G$ is the cardinality of the smallest ordered set of vertices that uniquely identifies every pair of distinct vertices (resp. edges) of $G$ by means of distance…
A graph is called uniquely distinguishing colorable if there is only one partition of vertices of the graph that forms distinguishing coloring with the smallest possible colors. In this paper, we study the unique colorability of the…
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…
An edge-colored connected graph $G$ is properly connected if between every pair of distinct vertices, there exists a path that no two adjacent edges have a same color. Fujita (2019) introduced the optimal proper connection number…
A graph G is (d_1,..,d_l)-colorable if the vertex set of G can be partitioned into subsets V_1,..,V_l such that the graph G[V_i] induced by the vertices of V_i has maximum degree at most d_i for all 1 <= i <= l. In this paper, we focus on…
We consider two possible extensions of a theorem of Thomassen characterizing the graphs admitting a 2-vertex-connected orientation. First, we show that the problem of deciding whether a mixed graph has a 2-vertex-connected orientation is…
The proper connection number $pc(G)$ of a connected graph $G$ is defined as the minimum number of colors needed to color its edges, so that every pair of distinct vertices of $G$ is connected by at least one path in $G$ such that no two…
A path in an edge-colored graph is called a proper path if no two adjacent edges of the path are colored with one same color. An edge-colored graph is called $k$-proper connected if any two vertices of the graph are connected by $k$…