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Register a new userThe rainbow connection number of enhanced power graph
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Let $G$ be a finite group, the enhanced power graph of $G$, denoted by $Gamma_G^e$, is the graph with vertex set $G$ and two vertices $x,y$ are edge connected in $Gamma_{G}^e$ if there exist $zin G$ such that $x,yinlangle zrangle$. Let $zeta$ be a edge-coloring of $Gamma_G^e$. In this article, we calculate the rainbow connection number of the enhanced power graph $Gamma_G^e$.
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Let $k$ be a positive integer, and $G$ be a $k$-connected graph. An edge-coloured path is emph{rainbow} if all of its edges have distinct colours. The emph{rainbow $k$-connection number} of $G$, denoted by $rc_k(G)$, is the minimum number of colours in an edge-colouring of $G$ such that, any two vertices are connected by $k$ internally vertex-disjoint rainbow paths. The function $rc_k(G)$ was introduced by Chartrand, Johns, McKeon and Zhang in 2009, and has since attracted significant interest. Let $t_k(n,r)$ denote the minimum number of edges in a $k$-connected graph $G$ on $n$ vertices with $rc_k(G)le r$. Let $s_k(n,r)$ denote the maximum number of edges in a $k$-connected graph $G$ on $n$ vertices with $rc_k(G)ge r$. The functions $t_1(n,r)$ and $s_1(n,r)$ have previously been studied by various authors. In this paper, we study the functions $t_2(n,r)$ and $s_2(n,r)$. We determine bounds for $t_2(n,r)$ which imply that $t_2(n,2)=(1+o(1))nlog_2 n$, and $t_2(n,r)$ is linear in $n$ for $rge 3$. We also provide some remarks about the function $s_2(n,r)$.
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 ${mathrm{pc}_{mathrm{opt}}}(G)$ has an upper bound in terms of the independence number $alpha(G)$. Namely, we prove that for a connected graph $G$, ${mathrm{pc}_{mathrm{opt}}}(G)le frac{5alpha(G)-1}{2}$. Moreoevr, for the case $alpha(G)leq 3$, we improve the upper bound to $4$, which is tight.
Let $F$ be a fixed graph. The rainbow Turan number of $F$ is defined as the maximum number of edges in a graph on $n$ vertices that has a proper edge-coloring with no rainbow copy of $F$ (where a rainbow copy of $F$ means a copy of $F$ all of whose edges have different colours). The systematic study of such problems was initiated by Keevash, Mubayi, Sudakov and Verstraete. In this paper, we show that the rainbow Turan number of a path with $k+1$ edges is less than $left(frac{9k}{7}+2right) n$, improving an earlier estimate of Johnston, Palmer and Sarkar.
An edge-cut $R$ of an edge-colored connected graph is called a rainbow-cut if no two edges in the edge-cut are colored the same. An edge-colored graph is rainbow disconnected if for any two distinct vertices $u$ and $v$ of the graph, there exists a $u$-$v$-rainbow-cut separating them. For a connected graph $G$, the rainbow disconnection number of $G$, denoted by rd$(G)$, is defined as the smallest number of colors that are needed in order to make $G$ rainbow disconnected. In this paper, we first give some tight upper bounds for rd$(G)$, and moreover, we completely characterize the graphs which meet the upper bound of the Nordhaus-Gaddum type results obtained early by us. Secondly, we propose a conjecture that $lambda^+(G)leq textnormal{rd}(G)leq lambda^+(G)+1$, where $lambda^+(G)$ is the upper edge-connectivity, and prove the conjecture for many classes of graphs, to support it. Finally, we give the relationship between rd$(G)$ of a graph $G$ and the rainbow vertex-disconnection number rvd$(L(G))$ of the line graph $L(G)$ of $G$.
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-colour
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