Surfaces, Tree-Width, Clique-Minors, and Partitions

  title={Surfaces, Tree-Width, Clique-Minors, and Partitions},
  author={Guoli Ding and Bogdan Oporowski and Daniel P. Sanders and Dirk L. Vertigan},
  journal={J. Comb. Theory, Ser. B},
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. 
Vertex Partitions of K4,4-Minor Free Graphs
It is proved that a 4-connected K4,4-minor free graph on n vertices has at most 4n−8 edges and this result is used to show that every K5,5,4 free graph has vertex-arboricity at least 4.
Some Recent Progress and Applications in Graph Minor Theory
In the core of the seminal Graph Minor Theory of Robertson and Seymour lies a powerful theorem capturing the ``rough'' structure of graphs excluding a fixed minor. This result was used to prove
Treewidth of Grid Subsets
A generalization of this claim is used to prove that the vertex set of Q_n cannot be partitioned to two parts, each of them inducing a subgraph of bounded tree-width.
Some results on cubic graphs
Pursuing a question of Oxley, we investigate whether the edge set of a graph admits a bipartition so that the contraction of either partite set produces a series-parallel graph. While Oxley’s
Planar Graphs and Partial k-Trees
It is proved that any Hamiltonian planar graph on n vertices can be decomposed into a forest and a graph of O(log n) treewidth, and provided an efficient algorithm for constructing this decomposition.
Bounds of spectral radii of K_{2,3}-minor free graphs
Let A(G) be the adjacency matrix of a graph G. The largest eigenvalue of A(G) is called spectral radius of G. In this paper, an upper bound of spectral radii of K2,3-minor free graphs with order n is
Tree-width, clique-minors, and eigenvalues
Planar Graphs have Bounded Queue-Number Vida Dujmović
We show that planar graphs have bounded queue-number, thus proving a conjecture of Heath, Leighton and Rosenberg from 1992. The key to the proof is a new structural tool called layered partitions,
Hadwiger's Conjecture
  • P. Seymour
  • Mathematics
    Open Problems in Mathematics
  • 2016
This is a survey of Hadwiger’s conjecture from 1943, that for all t ≥ 0, every graph either can be t-coloured, or has a subgraph that can be contracted to the complete graph on t + 1 vertices. This


Outerplanar Partitions of Planar Graphs
A conjecture about 4-connected maximal planar graphs that implies the original conjecture and a weaker form of the conjecture in which outerplanar sub graphs are replaced by subgraphs with no homeomorphs ofK4 is verified.
Graph Minors. II. Algorithmic Aspects of Tree-Width
Partitioning the Edges of a Planar Graph into Two Partial K-Trees
Two results are proved on partitioning the edges of a planar graph into two partial k-trees, for fixed values of k, and a recursive procedure is shown to construct an infinite family of planar graphs in which every member does not admit a partitioning into a partial 1-tree and a partial 2-tree.
Graph minors. III. Planar tree-width
Characterization and recognition of partial 3-trees
A set of confluent graph reductions is found such that any graph can be reduced to the empty graph if and only if it is a subgraph of a 3-tree.
On Hamilton cycles in certain planar graphs
Let G be a 2-connected plane graph with outer cycle XG such that for every minimal vertex cut S of G with |S| ≤ 3, every component of G\S contains a vertex of XG. A sufficient condition for G to be
The point-arboricity of a graph
The point-arboricity ρ(G) of a graphG is defined as the minimum number of subsets in a partition of the point set ofG so that each subset induces an acyclic subgraph. Dually, the tuleity τ(G) is the
Graph Coloring Problems
Planar Graphs. Graphs on Higher Surfaces. Degrees. Critical Graphs. The Conjectures of Hadwiger and Hajos. Sparse Graphs. Perfect Graphs. Geometric and Combinatorial Graphs. Algorithms.