17 research outputs found
Surfaces, Tree-Width, Clique-Minors, and Partitions
In 1971, Chartrand, Geller, and Hedetniemi conjectured that the edge set of a planar graph may be partitioned into two subsets, each of which induces an outerplanar graph. Some partial results towards this conjecture are presented. One such result, in which a planar graph may be thus edge partitioned into two series-parallel graphs, has nice generalizations for graphs embedded onto an arbitrary surface and graphs with no large clique-minor. Several open questions are raised. © 2000 Academic Press
Edge Partitions of Optimal -plane and -plane Graphs
A topological graph is a graph drawn in the plane. A topological graph is
-plane, , if each edge is crossed at most times. We study the
problem of partitioning the edges of a -plane graph such that each partite
set forms a graph with a simpler structure. While this problem has been studied
for , we focus on optimal -plane and -plane graphs, which are
-plane and -plane graphs with maximum density. We prove the following
results. (i) It is not possible to partition the edges of a simple optimal
-plane graph into a -plane graph and a forest, while (ii) an edge
partition formed by a -plane graph and two plane forests always exists and
can be computed in linear time. (iii) We describe efficient algorithms to
partition the edges of a simple optimal -plane graph into a -plane graph
and a plane graph with maximum vertex degree , or with maximum vertex
degree if the optimal -plane graph is such that its crossing-free edges
form a graph with no separating triangles. (iv) We exhibit an infinite family
of simple optimal -plane graphs such that in any edge partition composed of
a -plane graph and a plane graph, the plane graph has maximum vertex degree
at least and the -plane graph has maximum vertex degree at least .
(v) We show that every optimal -plane graph whose crossing-free edges form a
biconnected graph can be decomposed, in linear time, into a -plane graph and
two plane forests
The -index of graphs without intersecting triangles/quadrangles as a minor
The -matrix of a graph is the convex linear combination of
the adjacency matrix and the diagonal matrix of vertex degrees ,
i.e., , where . The -index of is the largest eigenvalue of .
Particularly, the matrix (resp. ) is exactly the
adjacency matrix (resp. signless Laplacian matrix) of . He, Li and Feng
[arXiv:2301.06008 (2023)] determined the extremal graphs with maximum adjacency
spectral radius among all graphs of sufficiently large order without
intersecting triangles and quadrangles as a minor, respectively. Motivated by
the above results of He, Li and Feng, in this paper we characterize the
extremal graphs with maximum -index among all graphs of sufficiently
large order without intersecting triangles and quadrangles as a minor for any
, respectively. As by-products, we determine the extremal graphs
with maximum signless Laplacian radius among all graphs of sufficiently large
order without intersecting triangles and quadrangles as a minor, respectively.Comment: 15 page
Bounding tree-width via contraction on the projective plane and torus
If X is a collection of edges in a graph G, let G/X denote the contraction of X. Following a question of Oxley and a conjecture of Oporowski, we prove that every projective-planar graph G admits an edge-partition {X,Y} such that G/X and G/Y have tree-width at most three. We prove that every toroidal graph G admits an edge-partition {X,Y} such that G/X and G/Y have tree-width at most three and four, respectively
Hamiltonicity and generalised total colourings of planar graphs
The total generalised colourings considered in this paper are colourings
of graphs such that the vertices and edges of the graph which receive the
same colour induce subgraphs from two prescribed hereditary graph properties while incident elements receive different colours. The associated total
chromatic number is the least number of colours with which this is possible.
We study such colourings for sets of planar graphs and determine, in
particular, upper bounds for these chromatic numbers for proper colourings
of the vertices while the monochromatic edge sets are allowed to be forests.
We also prove that if an even planar triangulation has a Hamilton cycle H
for which there is no cycle among the edges inside H, then such a graph
needs at most four colours for a total colouring as described above.
The paper is concluded with some conjectures and open problems.In part by the National Research
Foundation of South Africa (Grant Numbers 90841, 91128).http://www.degruyter.com/view/j/dmgtam2016Mathematics and Applied Mathematic
Product structure of graph classes with strongly sublinear separators
We investigate the product structure of hereditary graph classes admitting
strongly sublinear separators. We characterise such classes as subgraphs of the
strong product of a star and a complete graph of strongly sublinear size. In a
more precise result, we show that if any hereditary graph class
admits separators, then for any fixed
every -vertex graph in is a subgraph
of the strong product of a graph with bounded tree-depth and a complete
graph of size . This result holds with if
we allow to have tree-depth . Moreover, using extensions of
classical isoperimetric inequalties for grids graphs, we show the dependence on
in our results and the above bound are
both best possible. We prove that -vertex graphs of bounded treewidth are
subgraphs of the product of a graph with tree-depth and a complete graph of
size , which is best possible. Finally, we investigate the
conjecture that for any hereditary graph class that admits
separators, every -vertex graph in is a
subgraph of the strong product of a graph with bounded tree-width and a
complete graph of size . We prove this for various classes
of interest.Comment: v2: added bad news subsection; v3: removed section "Polynomial
Expansion Classes" which had an error, added section "Lower Bounds", and
added a new author; v4: minor revisions and corrections