On minimal triangle‐free 6‐chromatic graphs

All Ramsey (2K2,C4)−Minimal Graphs

Let F, G and H be non-empty graphs. The notation F → (G,H) means that if any edge of F is colored by red or blue, then either the red subgraph of F con- tains a graph G or the blue subgraph of F contains a graph H. A graph F (without isolated vertices) is called a Ramsey (G,H)−minimal if F → (G,H) and for every e ∈ E(F), (F − e) 9 (G,H). The set of all Ramsey (G,H)−minimal graphs is denoted by ...

minimal, vertex minimal and commonality minimal cn-dominating graphs

we define minimal cn-dominating graph $mathbf {mcn}(g)$, commonality minimal cn-dominating graph $mathbf {cmcn}(g)$ and vertex minimal cn-dominating graph $mathbf {m_{v}cn}(g)$, characterizations are given for graph $g$ for which the newly defined graphs are connected. further serval new results are developed relating to these graphs.

On Ramsey Minimal Graphs

On Ramsey Minimal Graphs

A graph G is r-ramsey-minimal with respect to Kk if every rcolouring of the edges of G yields a monochromatic copy of Kk, but the same is not true for any proper subgraph of G. In this paper we show that for any integer k ≥ 3 and r ≥ 2, there exists a constant c > 1 such that for large enough n, there exist at least c 2 non-isomorphic graphs on at most n vertices, each of which is r-ramsey-mini...

On Ramsey Minimal Graphs

An elementary probabilistic argument is presented which shows that for every forest F other than a matching, and every graph G containing a cycle, there exists an infinite number of graphs J such that J → (F,G) but if we delete from J any edge e the graph J − e obtained in this way does not have this property. Introduction. All graphs in this note are undirected graphs, without loops and multip...