Songling Shan

Homeomorphically irreducible spanning trees in locally connected graphs

A spanning tree T of a graph G is called a homeomorphically irreducible spanning tree (HIST) if T does not contain vertices of degree 2. A graph G is called locally connected if, for every vertex v ∈ V(G), the subgraph induced by the neighbourhood of v is connected. In this paper, we prove that every connected and locally connected graph with more than 3 vertices contains a HIST. Consequently, we confirm the following conjecture due to Archdeacon: every graph that triangulates some surface has a HIST, which was proposed as a question by Albertson, Berman, Hutchinson and Thomassen.

Nonempty intersection of longest paths in series-parallel graphs

In 1966 Gallai asked whether all longest paths in a connected graph have nonempty intersection. This is not true in general and various counterexamples have been found. However, the answer to Gallais question is positive for several well-known classes of graphs, as for instance connected outerplanar graphs, connected split graphs, and 2-trees. A graph is series-parallel if it does not contain K 4 as a minor. Series-parallel graphs are also known as partial 2-trees, which are arbitrary subgraphs of 2-trees. We present two independent proofs that every connected series-parallel graph has a vertex that is common to all of its longest paths. Since 2-trees are maximal series-parallel graphs, and outerplanar graphs are also series-parallel, our result captures these two classes in one proof and strengthens them to a larger class of graphs. We also describe how one such vertex can be found in linear time.

The Existence of a 2-Factor in a Graph Satisfying the Local Chvátal--Erdös Condition

The well-known Chvatal--Erdos theorem states that every graph

Homeomorphically irreducible spanning trees

G

On Equitable Vertex Distinguishing Edge Colorings of Trees

of order at least three with

Disjoint Chorded Cycles of the Same Length

\alpha(G)\le\kappa(G)

Spanning Trails with Maximum Degree at Most 4 in \(2K_2\)-Free Graphs

has a Hamiltonian cycle, where

Forbidden Pairs and the Existence of a Spanning Halin Subgraph

\alpha(G)

Characterizing Forbidden Pairs for Hamiltonian Squares

and

Chromatic index determined by fractional chromatic index

\kappa(G)