# A Note on Coloring $(4K_1, C_4, C_6)$-free graphs with a $C_7$

@inproceedings{Koutecky2021ANO, title={A Note on Coloring \$(4K\_1, C\_4, C\_6)\$-free graphs with a \$C\_7\$}, author={Martin Kouteck'y}, year={2021} }

Even-hole-free graphs are a graph class of much interest. Foley et al. [Graphs Comb. 36(1): 125-138 (2020)] have recently studied (4K1, C4, C6)-free graphs, which form a subclass of even-holefree graphs. Specifically, Foley et al. have shown an algorithm for coloring these graphs via bounded clique-width if they contain a C7. In this note, we give a simpler and much faster algorithm via a more restrictive graph parameter, neighborhood diversity.

## References

SHOWING 1-10 OF 13 REFERENCES

The Intersection of Two Vertex Coloring Problems

- Mathematics, Computer ScienceGraphs Comb.
- 2020

This paper presents partial results on the problem of coloring even hole-free graphs and describes the construction of a graph G that is L -free if G does not contain any graph in L as an induced subgraph.

Cliquewidth III: The Odd Case of Graph Coloring Parameterized by Cliquewidth

- Computer Science, MathematicsSODA
- 2018

GC is the first natural problem known to require exponential dependence on the parameter in the exponent of n and is closed by proving a lower bound of n2o(k), which shows that GC behaves qualitatively different from the other three problems.

Parameterized Algorithms for Modular-Width

- Mathematics, Computer ScienceIPEC
- 2013

It is argued that modular-width occupies a sweet spot as a graph parameter, generalizing several simpler notions on dense graphs but still evading the "price of generality" paid by clique-width.

A faster algorithm to recognize even-hole-free graphs

- Mathematics, Computer ScienceJ. Comb. Theory, Ser. B
- 2012

The problem of determining whether an n-node m-edge graph has an even hole, i.e., an induced simple cycle consisting of an even number of nodes, is solved in time O(n11).

Decomposition of even-hole-free graphs with star cutsets and 2-joins

- Computer Science, MathematicsJ. Comb. Theory, Ser. B
- 2013

Algorithmic Meta-theorems for Restrictions of Treewidth

- Mathematics, Computer ScienceAlgorithmica
- 2011

It is shown that any FO property in both of these classes with a singly exponential parameter dependence is possible and that it is possible to decide MSO logic on graphs of bounded vertex cover with a doubly exponentialParameter dependence, and is proved that the upper bound results cannot be improved significantly, under widely believed complexity assumptions.

Even-hole-free graphs part II: Recognition algorithm

- Mathematics, Computer ScienceJ. Graph Theory
- 2002

An algorithm that determines in polytime whether a graph contains an even hole is presented, based on a decomposition theorem for even-hole-free graphs obtained in Part I of this paper.

Even-hole-free graphs part I: Decomposition theorem

- Mathematics, Computer ScienceJ. Graph Theory
- 2002

A decomposition theorem for even-hole-free graphs is proved and this theorem is used in the second part of this paper to obtain a polytime recognition algorithm for even -hole- free graphs.

Evaluating and Tuning n-fold Integer Programming

- Computer Science, MathematicsSEA
- 2018

The original algorithm is practically unusable, but a series of improvements are discovered which make its evaluation possible and a new strategy for finding "approximatelly best" steps wildly outperforms the original construction.

An Algorithmic Theory of Integer Programming

- Mathematics, Computer ScienceArXiv
- 2019

It is shown that integer programming can be solved in time, and a strongly-polynomial algorithm is derived, that is, with running time $g(a,d)\textrm{poly}(n)$, independent of the rest of the input data.