The 3-rainbow index of a graph

Let $G$ be a nontrivial connected graph with an edge-coloring $c: E(G)\rightarrow \{1,2,...,q\},$ $q \in \mathbb{N}$, where adjacent edges may be colored the same. A tree $T$ in $G$ is a $rainbow tree$ if no two edges of $T$ receive the same color. For a vertex subset $S\subseteq V(G)$, a tree that connects $S$ in $G$ is called an $S$-tree. The minimum number of colors that are needed in an edge-coloring of $G$ such that there is a rainbow $S$-tree for each $k$-subset $S$ of $V(G)$ is called $k$-rainbow index, denoted by $rx_k(G)$. In this paper, we first determine the graphs whose 3-rainbow index equals 2, $m,$ $m-1$, $m-2$, respectively. We also obtain the exact values of $rx_3(G)$ for regular complete bipartite and multipartite graphs and wheel graphs. Finally, we give a sharp upper bound for $rx_3(G)$ of 2-connected graphs and 2-edge connected graphs, and graphs whose $rx_3(G)$ attains the upper bound are characterized.Comment: 13 page

A Study on Edge-Set Graphs of Certain Graphs

Let $G(V, E)$ be a simple connected graph, with $|E| = \epsilon.$ In this paper, we define an edge-set graph $\mathcal G_G$ constructed from the graph $G$ such that any vertex $v_{s,i}$ of $\mathcal G_G$ corresponds to the $i$-th $s$-element subset of $E(G)$ and any two vertices $v_{s,i}, v_{k,m}$ of $\mathcal G_G$ are adjacent if and only if there is at least one edge in the edge-subset corresponding to $v_{s,i}$ which is adjacent to at least one edge in the edge-subset corresponding to $v_{k,m}$ where $s,k$ are positive integers. It can be noted that the edge-set graph $\mathcal G_G$ of a graph $G$ id dependent on both the structure of $G$ as well as the number of edges $\epsilon.$ We also discuss the characteristics and properties of the edge-set graphs corresponding to certain standard graphs.Comment: 10 pages, 2 figure

Approximating k-Connected m-Dominating Sets

A subset $S$ of nodes in a graph $G$ is a $k$-connected $m$-dominating set ($(k,m)$-cds) if the subgraph $G[S]$ induced by $S$ is $k$-connected and every $v \in V \setminus S$ has at least $m$ neighbors in $S$. In the $k$-Connected $m$-Dominating Set ($(k,m)$-CDS) problem the goal is to find a minimum weight $(k,m)$-cds in a node-weighted graph. For $m \geq k$ we obtain the following approximation ratios. For general graphs our ratio $O(k \ln n)$ improves the previous best ratio $O(k^2 \ln n)$ and matches the best known ratio for unit weights. For unit disc graphs we improve the ratio $O(k \ln k)$ to $\min\left\{\frac{m}{m-k},k^{2/3}\right\} \cdot O(\ln^2 k)$ -- this is the first sublinear ratio for the problem, and the first polylogarithmic ratio $O(\ln^2 k)/\epsilon$ when $m \geq (1+\epsilon)k$; furthermore, we obtain ratio $\min\left\{\frac{m}{m-k},\sqrt{k}\right\} \cdot O(\ln^2 k)$ for uniform weights. These results are obtained by showing the same ratios for the Subset $k$-Connectivity problem when the set $T$ of terminals is an $m$-dominating set with $m \geq k$

Spanning Trees and Spanning Eulerian Subgraphs with Small Degrees. II

Let $G$ be a connected graph with $X\subseteq V(G)$ and with the spanning forest $F$. Let $\lambda\in [0,1]$ be a real number and let $\eta:X\rightarrow (\lambda,\infty)$ be a real function. In this paper, we show that if for all $S\subseteq X$, $\omega(G\setminus S)\le\sum_{v\in S}\big(\eta(v)-2\big)+2-\lambda(e_G(S)+1)$, then $G$ has a spanning tree $T$ containing $F$ such that for each vertex $v\in X$, $d_T(v)\le \lceil\eta(v)-\lambda\rceil+\max\{0,d_F(v)-1\}$, where $\omega(G\setminus S)$ denotes the number of components of $G\setminus S$ and $e_G(S)$ denotes the number of edges of $G$ with both ends in $S$. This is an improvement of several results and the condition is best possible. Next, we also investigate an extension for this result and deduce that every $k$-edge-connected graph $G$ has a spanning subgraph $H$ containing $m$ edge-disjoint spanning trees such that for each vertex $v$, $d_H(v)\le \big\lceil \frac{m}{k}(d_G(v)-2m)\big\rceil+2m$, where $k\ge 2m$; also if $G$ contains $k$ edge-disjoint spanning trees, then $H$ can be found such that for each vertex $v$, $d_H(v)\le \big\lceil \frac{m}{k}(d_G(v)-m)\big\rceil+m$, where $k\ge m$. Finally, we show that strongly $2$-tough graphs, including $(3+1/2)$-tough graphs of order at least three, have spanning Eulerian subgraphs whose degrees lie in the set $\{2,4\}$. In addition, we show that every $1$-tough graph has spanning closed walk meeting each vertex at most $2$ times and prove a long-standing conjecture due to Jackson and Wormald (1990).Comment: 46 pages, Keywords: Spanning tree; spanning Eulerian; spanning closed walk; connected factor; toughness; total exces