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تسجيل مستخدم جديدBounds for the rainbow disconnection number of graphs
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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 $G$.
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Let $G$ be a nontrivial edge-colored connected graph. An edge-cut $R$ of $G$ is called a {it rainbow edge-cut} if no two edges of $R$ are colored with the same color. For two distinct vertices $u$ and $v$ of $G$, if an edge-cut separates them, then t
he edge-cut is called a {it $u$-$v$-edge-cut}. An edge-colored graph $G$ is called emph{strong rainbow disconnected} if for every two distinct vertices $u$ and $v$ of $G$, there exists a both rainbow and minimum $u$-$v$-edge-cut ({it rainbow minimum $u$-$v$-edge-cut} for short) in $G$, separating them, and this edge-coloring is called a {it strong rainbow disconnection coloring} (srd-{it coloring} for short) of $G$. For a connected graph $G$, the emph{strong rainbow disconnection number} (srd-{it number} for short) of $G$, denoted by $textnormal{srd}(G)$, is the smallest number of colors that are needed in order to make $G$ strong rainbow disconnected. In this paper, we first characterize the graphs with $m$ edges such that $textnormal{srd}(G)=k$ for each $k in {1,2,m}$, respectively, and we also show that the srd-number of a nontrivial connected graph $G$ equals the maximum srd-number among the blocks of $G$. Secondly, we study the srd-numbers for the complete $k$-partite graphs, $k$-edge-connected $k$-regular graphs and grid graphs. Finally, we show that for a connected graph $G$, to compute $textnormal{srd}(G)$ is NP-hard. In particular, we show that it is already NP-complete to decide if $textnormal{srd}(G)=3$ for a connected cubic graph. Moreover, we show that for a given edge-colored (with an unbounded number of colors) connected graph $G$ it is NP-complete to decide whether $G$ is strong rainbow disconnected.
Let $G$ be a nontrivial connected and vertex-colored graph. A subset $X$ of the vertex set of $G$ is called rainbow if any two vertices in $X$ have distinct colors. The graph $G$ is called emph{rainbow vertex-disconnected} if for any two vertices $x$
and $y$ of $G$, there exists a vertex subset $S$ such that when $x$ and $y$ are nonadjacent, $S$ is rainbow and $x$ and $y$ belong to different components of $G-S$; whereas when $x$ and $y$ are adjacent, $S+x$ or $S+y$ is rainbow and $x$ and $y$ belong to different components of $(G-xy)-S$. Such a vertex subset $S$ is called a emph{rainbow vertex-cut} of $G$. For a connected graph $G$, the emph{rainbow vertex-disconnection number} of $G$, denoted by $rvd(G)$, is the minimum number of colors that are needed to make $G$ rainbow vertex-disconnected. In this paper, we obtain bounds of the rainbow vertex-disconnection number of a graph in terms of the minimum degree and maximum degree of the graph. We give a tighter upper bound for the maximum size of a graph $G$ with $rvd(G)=k$ for $kgeqfrac{n}{2}$. We then characterize the graphs of order $n$ with rainbow vertex-disconnection number $n-1$ and obtain the maximum size of a graph $G$ with $rvd(G)=n-1$. Moreover, we get a sharp threshold function for the property $rvd(G(n,p))=n$ and prove that almost all graphs $G$ have $rvd(G)=rvd(overline{G})=n$. Finally, we obtain some Nordhaus-Gaddum-type results: $n-5leq rvd(G)+rvd(overline{G})leq 2n$ and $n-1leq rvd(G)cdot rvd(overline{G})leq n^2$ for the rainbow vertex-disconnection numbers of nontrivial connected graphs $G$ and $overline{G}$ with order $ngeq 24$.
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 graph $G$ is called textit{rainbow} if every edge of $G$ receives a different color. Given any host graph $G$, the textit{anti-Ramsey} number of $t$ edge-disjoint rainbow spanning trees in $G$, denoted by $r(G,t)$, is defined as the m
aximum number of colors in an edge-coloring of $G$ containing no $t$ edge-disjoint rainbow spanning trees. For any vertex partition $P$, let $E(P,G)$ be the set of non-crossing edges in $G$ with respect to $P$. In this paper, we determine $r(G,t)$ for all host graphs $G$: $r(G,t)=|E(G)|$ if there exists a partition $P_0$ with $|E(G)|-|E(P_0,G)|<t(|P_0|-1)$; and $r(G,t)=max_{Pcolon |P|geq 3} {|E(P,G)|+t(|P|-2)}$ otherwise. As a corollary, we determine $r(K_{p,q},t)$ for all values of $p,q, t$, improving a result of Jia, Lu and Zhang.
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 e
dges 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.
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