Graphs with 3-rainbow index n−1n-1 and n−2n-2

Let $G$ be a nontrivial connected graph with an edge-coloring $c:E(G)\rightarrow \{1,2,\ldots,q\},$ $q\in \mathbb{N}$, 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 set $S\subseteq V(G)$, the tree connecting $S$ 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 $k$-set $S$ of $V(G)$ is called the $k$-rainbow index of $G$, denoted by $rx_k(G)$. In \cite{Zhang}, they got that the $k$-rainbow index of a tree is $n-1$ and the $k$-rainbow index of a unicyclic graph is $n-1$ or $n-2$. So there is an intriguing problem: Characterize graphs with the $k$-rainbow index $n-1$ and $n-2$. In this paper, we focus on $k=3$, and characterize the graphs whose 3-rainbow index is $n-1$ and $n-2$, respectively.Comment: 14 page

On Index Coding and Graph Homomorphism

In this work, we study the problem of index coding from graph homomorphism perspective. We show that the minimum broadcast rate of an index coding problem for different variations of the problem such as non-linear, scalar, and vector index code, can be upper bounded by the minimum broadcast rate of another index coding problem when there exists a homomorphism from the complement of the side information graph of the first problem to that of the second problem. As a result, we show that several upper bounds on scalar and vector index code problem are special cases of one of our main theorems. For the linear scalar index coding problem, it has been shown in [1] that the binary linear index of a graph is equal to a graph theoretical parameter called minrank of the graph. For undirected graphs, in [2] it is shown that $\mathrm{minrank}(G) = k$ if and only if there exists a homomorphism from $\bar{G}$ to a predefined graph $\bar{G}_k$. Combining these two results, it follows that for undirected graphs, all the digraphs with linear index of at most k coincide with the graphs $G$ for which there exists a homomorphism from $\bar{G}$ to $\bar{G}_k$. In this paper, we give a direct proof to this result that works for digraphs as well. We show how to use this classification result to generate lower bounds on scalar and vector index. In particular, we provide a lower bound for the scalar index of a digraph in terms of the chromatic number of its complement. Using our framework, we show that by changing the field size, linear index of a digraph can be at most increased by a factor that is independent from the number of the nodes.Comment: 5 pages, to appear in "IEEE Information Theory Workshop", 201