Partitioning 3-edge-colored complete equi-bipartite graphs by monochromatic trees under a color degree condition

The monochromatic tree partition number of an $r$-edge-colored graph $G$, denoted by $t_r(G)$, is the minimum integer $k$ such that whenever the edges of $G$ are colored with $r$ colors, the vertices of $G$ can be covered by at most $k$ vertex-disjoint monochromatic trees. In general, to determine this number is very difficult. For 2-edge-colored complete multipartite graph, Kaneko, Kano, and Suzuki gave the exact value of $t_2(K(n_1,n_2,...,n_k))$. In this paper, we prove that if $n\geq 3$, and K(n,n) is 3-edge-colored such that every vertex has color degree 3, then $t_3(K(n,n))=3.$Comment: 16 page

A sharp upper bound for the rainbow 2-connection number of 2-connected graphs

A path in an edge-colored graph is called {\em rainbow} if no two edges of it are colored the same. For an $\ell$-connected graph $G$ and an integer $k$ with $1\leq k\leq \ell$, the {\em rainbow $k$-connection number} $rc_k(G)$ of $G$ is defined to be the minimum number of colors required to color the edges of $G$ such that every two distinct vertices of $G$ are connected by at least $k$ internally disjoint rainbow paths. Fujita et. al. proposed a problem that what is the minimum constant $\alpha>0$ such that for all 2-connected graphs $G$ on $n$ vertices, we have $rc_2(G)\leq \alpha n$. In this paper, we prove that $\alpha=1$ and $rc_2(G)=n$ if and only if $G$ is a cycle of order $n$, settling down this problem.Comment: 8 page

Bicyclic graphs with maximal revised Szeged index

The revised Szeged index $Sz^*(G)$ is defined as $Sz^*(G)=\sum_{e=uv \in E}(n_u(e)+ n_0(e)/2)(n_v(e)+ n_0(e)/2),$ where $n_u(e)$ and $n_v(e)$ are, respectively, the number of vertices of $G$ lying closer to vertex $u$ than to vertex $v$ and the number of vertices of $G$ lying closer to vertex $v$ than to vertex $u$, and $n_0(e)$ is the number of vertices equidistant to $u$ and $v$. Hansen used the AutoGraphiX and made the following conjecture about the revised Szeged index for a connected bicyclic graph $G$ of order $n \geq 6$: Sz^*(G)\leq \{{array}{ll} (n^3+n^2-n-1)/4,& {if $n$ is odd}, (n^3+n^2-n)/4, & {if $n$ is even}. {array}. with equality if and only if $G$ is the graph obtained from the cycle $C_{n-1}$ by duplicating a single vertex. This paper is to give a confirmative proof to this conjecture.Comment: 7 page

Nordhaus-Gaddum-type theorem for the rainbow vertex-connection number of a graph

A vertex-colored graph $G$ is rainbow vertex-connected if any pair of distinct vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex-connection number of $G$, denoted by $rvc(G)$, is the minimum number of colors that are needed to make $G$ rainbow vertex-connected. In this paper we give a Nordhaus-Gaddum-type result of the rainbow vertex-connection number. We prove that when $G$ and $\bar{G}$ are both connected, then $2\leq rvc(G)+rvc(\bar{G})\leq n-1$. Examples are given to show that both the upper bound and the lower bound are best possible for all $n\geq 5$.Comment: 6 page

The (strong) rainbow connection numbers of Cayley graphs of Abelian groups

A path in an edge-colored graph $G$, where adjacent edges may have the same color, is called a rainbow path if no two edges of the path are colored the same. The rainbow connection number $rc(G)$ of $G$ is the minimum integer $i$ for which there exists an $i$-edge-coloring of $G$ such that every two distinct vertices of $G$ are connected by a rainbow path. The strong rainbow connection number $src(G)$ of $G$ is the minimum integer $i$ for which there exists an $i$-edge-coloring of $G$ such that every two distinct vertices $u$ and $v$ of $G$ are connected by a rainbow path of length $d(u,v)$. In this paper, we give upper and lower bounds of the (strong) rainbow connection Cayley graphs of Abelian groups. Moreover, we determine the (strong) rainbow connection numbers of some special cases.Comment: 12 page

The kk-proper index of graphs

A tree $T$ in an edge-colored graph is a \emph{proper tree} if any two adjacent edges of $T$ are colored with different colors. Let $G$ be a graph of order $n$ and $k$ be a fixed integer with $2\leq k\leq n$. For a vertex set $S\subseteq V(G)$, a tree containing the vertices of $S$ in $G$ is called an \emph{$S$-tree}. An edge-coloring of $G$ is called a \emph{$k$-proper coloring} if for every set $S$ of $k$ vertices in $G$, there exists a proper $S$-tree in $G$. The \emph{$k$-proper index} of a nontrivial connected graph $G$, denoted by $px_k(G)$, is the smallest number of colors needed in a $k$-proper coloring of $G$. In this paper, some simple observations about $px_k(G)$ for a nontrivial connected graph $G$ are stated. Meanwhile, the $k$-proper indices of some special graphs are determined, and for every pair of positive integers $a$, $b$ with $2\leq a\leq b$, a connected graph $G$ with $px_k(G)=a$ and $rx_k(G)=b$ is constructed for each integer $k$ with $3\leq k\leq n$. Also, the graphs with $k$-proper index $n-1$ and $n-2$ are respectively characterized.Comment: 12 page

On various (strong) rainbow connection numbers of graphs

An edge-coloured path is \emph{rainbow} if all the edges have distinct colours. For a connected graph $G$, the \emph{rainbow connection number} $rc(G)$ is the minimum number of colours in an edge-colouring of $G$ such that, any two vertices are connected by a rainbow path. Similarly, the \emph{strong rainbow connection number} $src(G)$ is the minimum number of colours in an edge-colouring of $G$ such that, any two vertices are connected by a rainbow geodesic (i.e., a path of shortest length). These two concepts of connectivity in graphs were introduced by Chartrand et al.~in 2008. Subsequently, vertex-coloured versions of both parameters, $rvc(G)$ and $srvc(G)$, and a total-coloured version of the rainbow connection number, $trc(G)$, were introduced. In this paper we introduce the strong total rainbow connection number $strc(G)$, which is the version of the strong rainbow connection number using total-colourings. Among our results, we will determine the strong total rainbow connection numbers of some special graphs. We will also compare the six parameters, by considering how close and how far apart they can be from one another. In particular, we will characterise all pairs of positive integers $a$ and $b$ such that, there exists a graph $G$ with $trc(G)=a$ and $strc(G)=b$, and similarly for the functions $rvc$ and $srvc$.Comment: 17 page

Rainbow connection number of dense graphs

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 needed in order to make $G$ rainbow connected. In this paper we show that $rc(G)\leq 3$, if $|E(G)|\geq {{n-2}\choose 2}+2$, and $rc(G)\leq 4$, if $|E(G)|\geq {{n-3}\choose 2}+3$. These bounds are sharp.Comment: 8 page

Tricyclic graphs with maximal revised Szeged index

The revised Szeged index of a graph $G$ is defined as $Sz^*(G)=\sum_{e=uv \in E}(n_u(e)+ n_0(e)/2)(n_v(e)+ n_0(e)/2),$ where $n_u(e)$ and $n_v(e)$ are, respectively, the number of vertices of $G$ lying closer to vertex $u$ than to vertex $v$ and the number of vertices of $G$ lying closer to vertex $v$ than to vertex $u$, and $n_0(e)$ is the number of vertices equidistant to $u$ and $v$. In this paper, we give an upper bound of the revised Szeged index for a connected tricyclic graph, and also characterize those graphs that achieve the upper bound.Comment: 14 pages. arXiv admin note: text overlap with arXiv:1104.212

The (revised) Szeged index and the Wiener index of a nonbipartite graph

Hansen et. al. used the computer programm AutoGraphiX to study the differences between the Szeged index $Sz(G)$ and the Wiener index $W(G)$, and between the revised Szeged index $Sz^*(G)$ and the Wiener index for a connected graph $G$. They conjectured that for a connected nonbipartite graph $G$ with $n \geq 5$ vertices and girth $g \geq 5,$ $Sz(G)-W(G) \geq 2n-5.$ Moreover, the bound is best possible as shown by the graph composed of a cycle on 5 vertices, $C_5$, and a tree $T$ on $n-4$ vertices sharing a single vertex. They also conjectured that for a connected nonbipartite graph $G$ with $n \geq 4$ vertices, $Sz^*(G)-W(G) \geq \frac{n^2+4n-6}{4}.$ Moreover, the bound is best possible as shown by the graph composed of a cycle on 3 vertices, $C_3$, and a tree $T$ on $n-3$ vertices sharing a single vertex. In this paper, we not only give confirmative proofs to these two conjectures but also characterize those graphs that achieve the two lower bounds.Comment: 12 pages. arXiv admin note: text overlap with arXiv:1210.646