53 research outputs found
Characterizing the forbidden pairs for graphs to be super-edge-connected
Let be a set of given connected graphs. A graph is said to
be -free if contains no as an induced subgraph for any
. The graph is super-edge-connected if each minimum
edge-cut isolates a vertex in . In this paper, except for some special
graphs, we characterize all forbidden subgraph sets such that
every -free is super-edge-connected for and
Proof a conjecture on connectivity keeping odd paths in k-connected bipartite graphs
Luo, Tian and Wu (2022) conjectured that for any tree with bipartition
and , every -connected bipartite graph with minimum degree at
least , where max, contains a tree such that
is still -connected. Note that when
the tree is the path with order . In this paper, we proved that every
-connected bipartite graph with minimum degree at least contains a path of order such that
remains -connected. This shows that the conjecture is true for paths with
odd order. And for paths with even order, the minimum degree bound in this
paper is the bound in the conjecture plus one
Extra Connectivity of Strong Product of Graphs
The - of a connected graph is
the minimum cardinality of a set of vertices, if it exists, whose deletion
makes disconnected and leaves each remaining component with more than
vertices, where is a non-negative integer. The of graphs and is the graph with vertex set , where two distinct vertices are adjacent in if and only if and or
and or and . In this paper, we give the - of
, where is a maximally connected -regular graph for . As a byproduct, we get -
conditional fault-diagnosability of under model
- β¦