Extremal Graphs Having No Stable Cutset

Extremal Graphs Having No Stable Cutsets

A stable cutset in a graph is a stable set whose deletion disconnects the graph. It was conjectured by Caro and proved by Chen and Yu that any graph with n vertices and at most 2n − 4 edges contains a stable cutset. The bound is tight, as we will show that all graphs with n vertices and 2n− 3 edges without stable cutset arise recursively glueing together triangles and triangular prisms along an...

Extremal graphs having no matching cuts

A graph G = (V, E) is matching immune if there is no matching cut in G. We show that for any matching immune graph G, |E| ≥ ⌈3(|V | − 1)/2⌉. This bound is tight, as we define operations that construct, from a given vertex, exactly the class of matching immune graphs that attain the bound.

Extremal Stable Graphs

Extremal P4-stable graphs

Call a graph G k-stable (with respect to some graph H) if, deleting any k edges of G, the remaining graph still contains H as a subgraph. For a fixed H, the minimum number of edges in a k-stable graph is denoted by S(k). We prove general bounds on S(k) and compute the exact value of the function S(k) for H = P4. The main result can be applied to extremal k-edge-hamiltonian hypergraphs.

Characterization of graphs having extremal Randić indices

The higher Randić index Rt (G) of a simple graph G is defined as Rt (G) = ∑ i1i2···it+1 1 √ δi1δi2 · · · δit+1 , where δi denotes the degree of the vertex i and i1i2 · · · it+1 runs over all paths of length t in G. In [J.A. Rodríguez, A spectral approach to the Randić index, Linear Algebra Appl. 400 (2005) 339–344], the lower and upper bound on R1(G) was determined in terms of a kind of Laplaci...