Note on minimally kk-rainbow connected graphs

An edge-colored graph $G$, where adjacent edges may have the same color, is {\it rainbow connected} if every two vertices of $G$ are connected by a path whose edge has distinct colors. A graph $G$ is {\it $k$-rainbow connected} if one can use $k$ colors to make $G$ rainbow connected. For integers $n$ and $d$ let $t(n,d)$ denote the minimum size (number of edges) in $k$-rainbow connected graphs of order $n$. Schiermeyer got some exact values and upper bounds for $t(n,d)$. However, he did not get a lower bound of $t(n,d)$ for $3\leq d<\lceil\frac{n}{2}\rceil$. In this paper, we improve his lower bound of $t(n,2)$, and get a lower bound of $t(n,d)$ for $3\leq d<\lceil\frac{n}{2}\rceil$.Comment: 8 page

Extremal Infinite Graph Theory

We survey various aspects of infinite extremal graph theory and prove several new results. The lead role play the parameters connectivity and degree. This includes the end degree. Many open problems are suggested.Comment: 41 pages, 16 figure

Graphs that do not contain a cycle with a node that has at least two neighbors on it

We recall several known results about minimally 2-connected graphs, and show that they all follow from a decomposition theorem. Starting from an analogy with critically 2-connected graphs, we give structural characterizations of the classes of graphs that do not contain as a subgraph and as an induced subgraph, a cycle with a node that has at least two neighbors on the cycle. From these characterizations we get polynomial time recognition algorithms for these classes, as well as polynomial time algorithms for vertex-coloring and edge-coloring

Properties of minimally tt-tough graphs

A graph $G$ is minimally $t$-tough if the toughness of $G$ is $t$ and the deletion of any edge from $G$ decreases the toughness. Kriesell conjectured that for every minimally $1$-tough graph the minimum degree $\delta(G)=2$. We show that in every minimally $1$-tough graph $\delta(G)\le\frac{n+2}{3}$. We also prove that every minimally $1$-tough claw-free graph is a cycle. On the other hand, we show that for every $t \in \mathbb{Q}$ any graph can be embedded as an induced subgraph into a minimally $t$-tough graph