On the divisibility of graphs

@article{Hong2002OnTD,
  title={On the divisibility of graphs},
  author={Ch{\'i}nh T. Ho{\`a}ng and Colin McDiarmid},
  journal={Discret. Math.},
  year={2002},
  volume={242},
  pages={145-156}
}
Some problems on induced subgraphs
Coloring graphs with no even holes ≥ 6: the triangle-free case
TLDR
It is proved that the class of graphs with no triangle and no induced cycle of even length at least 6 has bounded chromatic number and the existence of C_4 is allowed.
On the chromatic number of a family of odd hole free graphs
TLDR
It is proved that for (odd hole, full house)-free graph G, χ(G) ≤ ω(G)+ 1, and the equality holds if and only if ω (G) = 3 and G has H as an induced subgraph.
Clique-coloring some classes of odd-hole-free graphs
We consider the problem of clique-coloring, that is coloring the vertices of a given graph such that no maximal clique of size at least 2 is monocolored. Whereas we do not know any odd-hole-free
Vertex Colouring and Forbidden Subgraphs – A Survey
TLDR
This work surveys results on vertex colourings of graphs defined in terms of forbidden induced subgraph conditions in order to obtain useful results from a graph coloring formulation of his problem.
Four NP-complete problems about generalizations of perfect graphs
We show that the following problems are NP-complete. 1. Can the vertex set of a graph be partitioned into two sets such that each set induces a perfect graph? 2. Is the difference between the
Clique-Coloring Claw-Free Graphs
TLDR
It is proved that a claw-free graph with maximum degree at most 7, except an odd cycle longer than 3, has a 2-clique-coloring by using the decomposition theorem of Chudnovsky and Seymour.
A χ-binding function for the class of { 3 K 1 , K 1 ∪ K 4 }-free graphs
We prove that the chromatic number of any {3K1,K1∪K4}-free graph is at most a factor 28/15 times its clique number. In order to prove this result we prove that any connected subcubic triangle-free
K4-free graphs with no odd holes
...
...

References

SHOWING 1-10 OF 12 REFERENCES
The strong perfect-graph conjecture is true for K1, 3-free graphs
Colouring graphs with prescribed induced cycle lengths
TLDR
The surprising result is obtained that there exists no linear ´-binding function for G I (3;4), the class of all graphs whose induced cycle lengths are 4 or 5.
On the NP-completeness of the k-colorability problem for triangle-free graphs
The Ramsey Number R(3, t) Has Order of Magnitude t2/log t
TLDR
It is proved that R(3, t) is bounded below by (1 – o(1))t/2/log t times a positive constant, and it follows that R (3), the Ramsey number for positive integers s and t, has asymptotic order of magnitude t2/ log t.
Random graphs
  • A. Rucinski
  • Mathematics
    ZOR Methods Model. Oper. Res.
  • 1989
TLDR
I study random graphs as a probabilist dealing with some combinatorial structures, and my methods are probabilistic and based on analysis, using for example integration theory, functional analysis, martingales and stochastic integration.
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.
Paw-Fee Graphs
Colouring random graphs
  • C. McDiarmid
  • Mathematics, Computer Science
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 1975
TLDR
This work discusses some results concerned with the behaviour of colouring algorithms on large random graphs and investigates the role of noise in the choice of colours.
...
...