39 research outputs found
Joint-tree model and the maximum genus of graphs
The vertex v of a graph G is called a 1-critical-vertex for the maximum genus
of the graph, or for simplicity called 1-critical-vertex, if G-v is a connected
graph and {\deg}M(G - v) = {\deg}M(G) - 1. In this paper, through the
joint-tree model, we obtained some types of 1-critical-vertex, and get the
upper embeddability of the Spiral Snm
The reducibility of optimal 1-planar graphs with respect to the lexicographic product
A graph is called 1-planar if it can be drawn on the plane (or on the sphere)
such that each edge is crossed at most once. A 1-planar graph is called
optimal if it satisfies . If and are graphs, then
the lexicographic product has vertex set the Cartesian product
and edge set . A graph is
called reducible if it can be expressed as the lexicographic product of two
smaller non-trivial graphs. In this paper, we prove that an optimal 1-planar
graph is reducible if and only if is isomorphic to the complete
multipartite graph . As a corollary, we prove that every reducible
1-planar graph with vertices has at most edges for or . We also prove that this bound is tight for infinitely many values of .
Additionally, we give two necessary conditions for a graph to be
1-planar.Comment: 23 pages, 14 fugure
The maximum genus of graphs with diameter three
AbstractThis paper shows that if G is a simple graph with diameter three then G is up-embeddable unless G is either a Ξ2-graph (Fig. 1) or a Ξ3-graph (Fig. 2) with ΞΎ(G) = 2, i.e., the maximum genus Ξ³M(G) = (Ξ²(G) β 2)/2
On the sizes of generalized cactus graphs
A cactus is a connected graph in which each edge is contained in at most one
cycle. We generalize the concept of cactus graphs, i.e., a -cactus is a
connected graph in which each edge is contained in at most cycles where
. It is well known that every cactus with vertices has at most
edges. Inspired by it, we attempt to
establish analogous upper bounds for general -cactus graphs. In this paper,
we first characterize -cactus graphs for based on the block
decompositions. Subsequently, we give tight upper bounds on their sizes.
Moreover, the corresponding extremal graphs are also characterized. However,
the case of remains open. For the case of 2-connectedness, the range
of is expanded to all positive integers in our research. We prove that
every -connected -cactus graphs with vertices has at most
edges, and the bound is tight if . But, for ,
determining best bounds remains a mystery except for some small values of .Comment: 14 pages, 2 figure
The Crossing Number of the Circulant Graph C
A graph G = (V,E) is a set V of vertices and a subset E of unordered pairs of vertices, called edges. A Smarandache drawing of a graph G is a drawing of G on the plane with minimal intersections for its each component
Vertex Splitting and Upper Embeddable Graphs
The weak minor G of a graph G is the graph obtained from G by a sequence of
edge-contraction operations on G. A weak-minor-closed family of upper
embeddable graphs is a set G of upper embeddable graphs that for each graph G
in G, every weak minor of G is also in G. Up to now, there are few results
providing the necessary and sufficient conditions for characterizing upper
embeddability of graphs. In this paper, we studied the relation between the
vertex splitting operation and the upper embeddability of graphs; provided not
only a necessary and sufficient condition for characterizing upper
embeddability of graphs, but also a way to construct weak-minor-closed family
of upper embeddable graphs from the bouquet of circles; extended a result in J:
Graph Theory obtained by L. Nebesk{\P}y. In addition, the algorithm complex of
determining the upper embeddability of a graph can be reduced much by the
results obtained in this paper
The generalized 4-connectivity of burnt pancake graphs
The generalized -connectivity of a graph , denoted by , is
the minimum number of internally edge disjoint -trees for any and . The generalized -connectivity is a natural extension of
the classical connectivity and plays a key role in applications related to the
modern interconnection networks. An -dimensional burnt pancake graph
is a Cayley graph which posses many desirable properties. In this paper, we try
to evaluate the reliability of by investigating its generalized
4-connectivity. By introducing the notation of inclusive tree and by studying
structural properties of , we show that for , that is, for any four vertices in , there exist () internally
edge disjoint trees connecting them in