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 $cfc(G)\le α(G)$ 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 $Δ(T)\ge \frac{α(T)+2}{2}$, then $cfc(T)=Δ(T)$.
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 Erdös-Gallai-type results for the conflict-free connection numbers of graphs.
A path in a vertex-colored graph is called {\it conflict-free} if there is a color used on exactly one of its vertices. A vertex-colored graph is said to be {\it conflict-free vertex-connected} if any two vertices of the graph are connected by a conflict-free path. The {\it conflict-free vertex-connection number}, denoted by $vcfc(G)$, is defined as the smallest number of colors required to make $G$ conflict-free vertex-connected. Li et al. conjectured that for a connected graph $G$ of order $n$, $vcfc(G)\leq vcfc(P_n)$. We confirm that the conjecture is true and pose a a relevant conjecture c…
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 $T$ …
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 Erdős-Rényi random graph model, in which each of the $\binom{n}{2}$ pairs of vertices appears as an edge with probability $p$ independent fr…
A path in a vertex-colored graph is called \emph{conflict free} if there is a color used on exactly one of its vertices. A vertex-colored graph is said to be \emph{conflict-free vertex-connected} if any two vertices of the graph are connected by a conflict-free path. This paper investigates the question: For a connected graph $G$, what is the smallest number of colors needed in a vertex-coloring of $G$ in order to make $G$ conflict-free vertex-connected. As a result, we get that the answer is easy for $2$-connected graphs, and very difficult for connected graphs with more cut-vertices, includi…
A path in an edge-colored graph is called \emph{conflict-free} if it contains at least one color used on exactly one of its edges. An edge-colored graph $G$ is \emph{conflict-free connected} if for any two distinct vertices of $G$, there is a conflict-free path connecting them. For a connected graph $G$, the \emph{conflict-free connection number} of $G$, denoted by $cfc(G)$, is defined as the minimum number of colors that are required to make $G$ conflict-free connected. In this paper, we investigate the conflict-free connection numbers of connected claw-free graphs, especially line graphs. We…
Let $G$ be an edge-colored connected graph. A path of $G$ is called rainbow if its every edge is colored by a distinct color. $G$ is called rainbow connected if there exists a rainbow path between every two vertices of $G$. The minimum number of colors that are needed to make $G$ rainbow connected is called the rainbow connection number of $G$, denoted by $rc(G)$. In this paper, we investigate the relation between the rainbow connection number and the independence number of a graph. We show that if $G$ is a connected graph, then $rc(G)\leq 2α(G)-1$. Two examples $G$ are given to show that the …
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{s…
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 ${\mathrm{pc}_{\mathrm{opt}}}(G)$ for a monochromatic connected graph $G$, to make a connected graph properly connected efficiently. More precisely, ${\mathrm{pc}_{\mathrm{opt}}}(G)$ is the smallest integer $p+q$ when one converts a given monochromatic graph $G$ into a properly connected graph by recoloring $p$ edges with $q$ colors. In this paper, we show that ${\mat…
An edge-colored graph $G$ is \emph{conflict-free connected} if, between each pair of distinct vertices, there exists a path containing a color used on exactly one of its edges. The \emph{conflict-free connection number} of a connected graph $G$, denoted by $cfc(G)$, is defined as the smallest number of colors that are needed in order to make $G$ conflict-free connected. In this paper, we determine all trees $T$ of order $n$ for which $cfc(T)=n-t$, where $t\geq 1$ and $n\geq 2t+2 $. Then we prove that $1\leq cfc(G)\leq n-1$ for a connected graph $G$, and characterize the graphs $G$ with $cfc(G)…
We study the conflict-free connection coloring of trees, which is also the conflict-free coloring of the so-called edge-path hypergraphs of trees. We first prove that for a tree $T$ of order $n$, $cfc(T)\geq cfc(P_n)=\lceil \log_{2} n\rceil$, which completely confirms the conjecture of Li and Wu. We then present a sharp upper bound for the conflict-free connection number of trees by a simple algorithm. Furthermore, we show that the conflict-free connection number of the binomial tree with $2^{k-1}$ vertices is $k-1$. At last, we study trees which are $cfc$-critical, and prove that if a tree $T…
An edge-colored graph $G$ is \emph{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 \emph{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. For a graph $G,$ let $C(G)$ be the subgraph of $G$ induced by its set of cut-edges. In this paper, we first show that, if $G$ is a connected non-complete graph $G$ of order $n\geq 9$ with $C(G)$ being a linear forest and with the minimum degree %$δ(G)\geq…
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 compu…
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 c…
Let G be a graph on n vertices in which every induced subgraph on s=\log^3 n vertices has an independent set of size at least t=\log n. What is the largest q=q(n) so that every such G must contain an independent set of size at least q ? This is one of several related questions raised by Erdos and Hajnal. We show that q(n)=Θ(\log^2 n/\log \log n), investigate the more general problem obtained by changing the parameters s and t, and discuss the connection to a related Ramsey-type problem.
The independence complex of a graph G is a simplicial complex whose simplices are the independent sets in G. In the last couple of decades, the independence complexes of square grids (with various boundary conditions) have gained much attention because of their connections with the hard square model from statistical physics. In this article, we prove that if G is an $m\times n$ grid with open or cylindrical boundary condition then its independence complex is homotopy equivalent to a wedge of spheres. A part of this result settles a conjecture of Iriye.
An edge-colored graph $G$ is called properly colored if no two adjacent edges share a color in $G$. An edge-colored connected graph $G$ is called properly connected if between every pair of distinct vertices, there exists a path that is properly colored. In this paper, we discuss how to make a connected graph properly connected efficiently. More precisely, we consider the problem to convert a given monochromatic graph into properly connected by recoloring $p$ edges with $q$ colors so that $p+q$ is as small as possible. We discuss how this can be done efficiently for some restricted graphs, suc…
A vertex-coloring of a hypergraph is conflict-free, if each edge contains a vertex whose color is not repeated on any other vertex of that edge. Let $f(r, Δ)$ be the smallest integer $k$ such that each $r$-uniform hypergraph of maximum vertex degree $Δ$ has a conflict-free coloring with at most $k$ colors. As shown by Tardos and Pach, similarly to a classical Brooks' type theorem for hypergraphs, $f(r, Δ)\leq Δ+1$. Compared to Brooks' theorem, according to which there is only a couple of graphs/hypergraphs that attain the $Δ+1$ bound, we show that there are several infinite classes of uniform …
The Birkhoff graph $\mathcal{B}_n$ is the Cayley graph of the symmetric group $S_n$, where two permutations are adjacent if they differ by a single cycle. Our main result is a tighter upper bound on the independence number $α(\mathcal{B}_n)$ of $\mathcal{B}_n$, namely, we show that $α(\mathcal{B}_n) \le O(n!/1.97^n)$ improving on the previous known bound of $α(\mathcal{B}_n) \le O(n!/\sqrt{2}^{n})$ by [Kane-Lovett-Rao, FOCS 2017]. Our approach combines a higher-order version of their representation theoretic techniques with linear programming. With an explicit construction, we also improve the…
