A generalization of heterochromatic graphs

In 2006, Suzuki, and Akbari & Alipour independently presented a necessary and sufficient condition for edge-colored graphs to have a heterochromatic spanning tree, where a heterochromatic spanning tree is a spanning tree whose edges have distinct colors. In this paper, we propose $f$-chromatic graphs as a generalization of heterochromatic graphs. An edge-colored graph is $f$-chromatic if each color $c$ appears on at most $f(c)$ edges. We also present a necessary and sufficient condition for edge-colored graphs to have an $f$-chromatic spanning forest with exactly $m$ components. Moreover, using this criterion, we show that a $g$-chromatic graph $G$ of order $n$ with $|E(G)|>\binom{n-m}{2}$ has an $f$-chromatic spanning forest with exactly $m$ ($1 \le m \le n-1$) components if $g(c) \le \frac{|E(G)|}{n-m}f(c)$ for any color $c$.Comment: 14 pages, 4 figure

The mincut graph of a graph

In this paper we introduce an intersection graph of a graph $G$, with vertex set the minimum edge-cuts of $G$. We find the minimum cut-set graphs of some well-known families of graphs and show that every graph is a minimum cut-set graph, henceforth called a \emph{mincut graph}. Furthermore, we show that non-isomorphic graphs can have isomorphic mincut graphs and ask the question whether there are sufficient conditions for two graphs to have isomorphic mincut graphs. We introduce the $r$-intersection number of a graph $G$, the smallest number of elements we need in $S$ in order to have a family $F=\{S_1, S_2 \ldots , S_i\}$ of subsets, such that $|S_i|=r$ for each subset. Finally we investigate the effect of certain graph operations on the mincut graphs of some families of graphs

Sufficient conditions for certain structural properties of graphs based on Wiener-type indices

Let $G=(V,E)$ be a simple connected graph with the vertex set $V$and the edge set $E$. The Wiener-type invariants of $G=(V,E)$ can beexpressed in terms of the quantities W_{f}=\sum_{\{u,v\}\subseteqV}f(d_{G}(u,v)) for various choices of the function $f$, where$d_{G}(u,v)$ is the distance between vertices $u$ and $v$ in $G$. Inthis paper, we establish sufficient conditions based on Wiener-typeindices under which every path of length $r$ is contained in aHamiltonian cycle, and under which a bipartite graph on $n+m$(m>n) vertices contains a cycle of size $2n$