Erd\"os-Gallai-type results for conflict-free connection of graphs

A path in an edge-colored graph is called \emph{a conflict-free path} if there exists a color used on only one of its edges. An edge-colored graph is called \emph{conflict-free connected} if there is a conflict-free path between each pair of distinct vertices. The \emph{conflict-free connection number} of a connected graph $G$, denoted by $\mathit{cfc}(G)$, is defined as the smallest number of colors that are required to make $G$ conflict-free connected. In this paper, we obtain Erd\"{o}s-Gallai-type results for the conflict-free connection numbers of graphs.Comment: 6 page

Conflict-free connection number of random graphs

An edge-colored graph $G$ is conflict-free connected if any two of its vertices are connected by a path which contains a color used on exactly one of its edges. The conflict-free connection number of a connected graph $G$, denoted by $cfc(G)$, is the smallest number of colors needed in order to make $G$ conflict-free connected. In this paper, we show that almost all graphs have the conflict-free connection number 2. More precisely, let $G(n,p)$ denote the Erd\H{o}s-R\'{e}nyi random graph model, in which each of the $\binom{n}{2}$ pairs of vertices appears as an edge with probability $p$ independent from other pairs. We prove that for sufficiently large $n$, $cfc(G(n,p))\le 2$ if $p\ge\frac{\log n +\alpha(n)}{n}$, where $\alpha(n)\rightarrow \infty$. This means that as soon as $G(n,p)$ becomes connected with high probability, $cfc(G(n,p))\le 2$.Comment: 13 page

Conflict-free connections: algorithm and complexity

A path in an(a) edge(vertex)-colored graph is called \emph{a conflict-free path} if there exists a color used on only one of its edges(vertices). An(A) edge(vertex)-colored graph is called \emph{conflict-free (vertex-)connected} if there is a conflict-free path between each pair of distinct vertices. We call the graph $G$ \emph{strongly conflict-free connected }if there exists a conflict-free path of length $d_G(u,v)$ for every two vertices $u,v\in V(G)$. And the \emph{strong conflict-free connection number} of a connected graph $G$, denoted by $scfc(G)$, is defined as the smallest number of colors that are required to make $G$ strongly conflict-free connected. In this paper, we first investigate the question: Given a connected graph $G$ and a coloring $c: E(or\ V)\rightarrow \{1,2,\cdots,k\} \ (k\geq 1)$of the graph, determine whether or not$G$is, respectively, conflict-free connected, vertex-conflict-free connected, strongly conflict-free connected under coloring$c$. We solve this question by providing polynomial-time algorithms. We then show that it is NP-complete to decide whether there is a k-edge-coloring$(k\geq 2)$of$G$such that all pairs$(u,v)\in P \ (P\subset V\times V)$are strongly conflict-free connected. Finally, we prove that the problem of deciding whether$scfc(G)\leq k$$(k\geq 2)$for a given graph$G$is NP-complete.Comment: 17 pages. The main change is in Subsection 3.2, Theorem 3.4, where we add the result and proof of the NP-completeness for the case$k=2$, which was not done in the old versio

Conflict-free (vertex)-connection numbers of graphs with small diameters

A path in an(a) edge(vertex)-colored graph is called a conflict-free path if there exists a color used on only one of its edges(vertices). An(A) edge(vertex)-colored graph is called conflict-free (vertex-)connected if for each pair of distinct vertices, there is a conflict-free path connecting them. For a connected graph $G$, the conflict-free (vertex-)connection number of $G$, denoted by $cfc(G)(\text{or}~vcfc(G))$, is defined as the smallest number of colors that are required to make $G$ conflict-free (vertex-)connected. In this paper, we first give the exact value $cfc(T)$ for any tree $T$ with diameters $2,3$ and $4$. Based on this result, the conflict-free connection number is determined for any graph $G$ with $diam(G)\leq 4$ except for those graphs $G$ with diameter $4$ and $h(G)=2$. In this case, we give some graphs with conflict-free connection number $2$ and $3$, respectively. For the conflict-free vertex-connection number, the exact value $vcfc(G)$ is determined for any graph $G$ with $diam(G)\leq 4$.Comment: 12 page

Conflict-free connection number and independence number of a graph

An edge-colored graph $G$ is conflict-free connected if any two of its vertices are connected by a path, which contains a color used on exactly one of its edges. The conflict-free connection number of a connected graph $G$, denoted by $cfc(G)$, is defined as the minimum number of colors that are required in order to make $G$ conflict-free connected. In this paper, we investigate the relation between the conflict-free connection number and the independence number of a graph. We firstly show that $cfc(G)\le \alpha(G)$ for any connected graph $G$, and an example is given showing that the bound is sharp. With this result, we prove that if $T$ is a tree with $\Delta(T)\ge \frac{\alpha(T)+2}{2}$, then $cfc(T)=\Delta(T)$

Graph colorings under global structural conditions

More than ten years ago in 2008, a new kind of graph coloring appeared in graph theory, which is the {\it rainbow connection coloring} of graphs, and then followed by some other new concepts of graph colorings, such as {\it proper connection coloring, monochromatic connection coloring, and conflict-free connection coloring} of graphs. In about ten years of our consistent study, we found that these new concepts of graph colorings are actually quite different from the classic graph colorings. These {\it colored connection colorings} of graphs are brand-new colorings and they need to take care of global structural properties (for example, connectivity) of a graph under the colorings; while the traditional colorings of graphs are colorings under which only local structural properties (adjacent vertices or edges) of a graph are taken care of. Both classic colorings and the new colored connection colorings can produce the so-called chromatic numbers. We call the colored connection numbers the {\it global chromatic numbers}, and the classic or traditional chromatic numbers the {\it local chromatic numbers}. This paper intends to clarify the difference between the colored connection colorings and the traditional colorings, and finally to propose the new concepts of global colorings under which global structural properties of the colored graph are kept, and the global chromatic numbers.Comment: 14 page

Monochromatic disconnection of graphs

For an edge-colored graph $G$, we call an edge-cut $M$ of $G$ monochromatic if the edges of $M$ are colored with a same color. The graph $G$ is called monochromatically disconnected if any two distinct vertices of $G$ are separated by a monochromatic edge-cut. For a connected graph $G$, the monochromatic disconnection number, denoted by $md(G)$, of $G$ is the maximum number of colors that are needed in order to make $G$ monochromatically disconnected. We will show that almost all graphs have monochromatic disconnection numbers equal to 1. We also obtain the Nordhaus-Gaddum-type results for $md(G)$.Comment: 16 page

Strong conflict-free connection of graphs

A path $P$ in an edge-colored graph is called \emph{a conflict-free path} if there exists a color used on only one of the edges of $P$. An edge-colored graph $G$ is called \emph{conflict-free connected} if for each pair of distinct vertices of $G$ there is a conflict-free path in $G$ connecting them. The graph $G$ is called \emph{strongly conflict-free connected }if for every pair of vertices $u$ and $v$ of $G$ there exists a conflict-free path of length $d_G(u,v)$ in $G$ connecting them. For a connected graph $G$, the \emph{strong conflict-free connection number} of $G$, denoted by $\mathit{scfc}(G)$, is defined as the smallest number of colors that are required in order to make $G$ strongly conflict-free connected. In this paper, we first show that if $G_t$ is a connected graph with $m$ $(m\geq 2)$ edges and $t$ edge-disjoint triangles, then $\mathit{scfc}(G_t)\leq m-2t$, and the equality holds if and only if $G_t\cong S_{m,t}$. Then we characterize the graphs $G$ with $scfc(G)=k$ for $k\in \{1,m-3,m-2,m-1,m\}$. In the end, we present a complete characterization for the cubic graphs $G$ with $scfc(G)=2$.Comment: 23 pages, 10 figure

Improved lower bounds on parity vertex colourings of binary trees

A vertex colouring is called a \emph{parity vertex colouring} if every path in $G$ contains an odd number of occurrences of some colour. Let $\chi_{p}(G)$ be the minimal number of colours in a parity vertex colouring of $G$. We show that $\chi_{p}(B^*) \ge \sqrt{d} + \frac{1}{4} \log_2(d) - \frac{1}{2}$ where $B^*$ is a subdivision of the complete binary tree $B_d$. This improves the previously known bound $\chi_{p}(B^*) \ge \sqrt{d}$ and enhances the techniques used for proving lower bounds. We use this result to show that $\chi_{p}(T) > \sqrt[3]{\log{n}}$ where $T$ is any binary tree with $n$ vertices. These lower bounds are also lower bounds for the conflict-free colouring. We also prove that $\chi_{p}(G)$ is not monotone with respect to minors and determine its value for cycles. Furthermore, we study complexity of computing the parity vertex chromatic number $\chi_{p}(G)$. We show that checking whether a vertex colouring is a parity vertex colouring is coNP-complete. Then we use Courcelle's theorem to prove that the problem of checking whether $\chi_{p}(G) \le k$ is fixed-parameter tractable with respect $k$ and the treewidth of $G$

Hardness results for three kinds of colored connections of graphs

The concept of rainbow connection number of a graph was introduced by Chartrand et al. in 2008. Inspired by this concept, other concepts on colored version of connectivity in graphs were introduced, such as the monochromatic connection number by Caro and Yuster in 2011, the proper connection number by Borozan et al. in 2012, and the conflict-free connection number by Czap et al. in 2018, as well as some other variants of connection numbers later on. Chakraborty et al. proved that to compute the rainbow connection number of a graph is NP-hard. For a long time, it has been tried to fix the computational complexity for the monochromatic connection number, the proper connection number and the conflict-free connection number of a graph. However, it has not been solved yet. Only the complexity results for the strong version, i.e., the strong proper connection number and the strong conflict-free connection number, of these connection numbers were determined to be NP-hard. In this paper, we prove that to compute each of the monochromatic connection number, the proper connection number and the conflict free connection number for a graph is NP-hard. This solves a long standing problem in this field, asked in many talks of workshops and papers.Comment: 23 pages, 7 figure