The 2-Factor Polynomial Detects Even Perfect Matchings
- MathematicsElectron. J. Comb.
It is shown that the $2-factor polynomial, an invariant of a planar trivalent graph with a perfect matching, counts the number of $2$-factors that contain the perfect matching as a subgraph and detects even perfect matchings.
Clustered coloring of graphs excluding a subgraph
- Mathematics, Computer Science
This paper studies clustered coloring, where the number of colors depends on an excluded subgraph, a much weaker assumption than previous works, and proves the following: graphs of bounded treewidth and with no no-subgraph are s+1-choosable with bounded clustering, which is best possible.
Separating sets of strings by finding matching patterns is almost always hard
- Computer ScienceTheor. Comput. Sci.
Finding Patterns is Almost Always Hard
- Computer ScienceArXiv
It is shown that parameterized analysis of the problem of finding a small set of patterns that match one set of strings but do not match any string in a second set is difficult, and that parameterizing by the size of pattern set and the number of strings gives FPT results, amongst others.
Filling the complexity gaps for colouring planar and bounded degree graphs
Using known examples of non-3-choosable and non-4-choOSable graphs, the complexity of $k-Regular List Colouring restricted to planar graphs is classified and a number of related colouring problems for graphs with bounded maximum degree are classified.
The Four Color Algorithm
This paper proposes an algorithm that proves an NP-complete 4-color theorem by employing a linear time complexity where . The proposed algorithm accurately halves the vertex set V of the graph $G
Comparison of triple-patterning decomposition algorithms using aperiodic tiling patterns
- Computer SciencePhotomask Japan
This paper compares the scalability of different coloring algorithms using a variety of contact patterns based on Penrose Tiles, proving that a set of aperiodic tiling known as "Penrose Tiling" is 3-colorable.
The Four-Colour Theorem
- MathematicsJ. Comb. Theory, Ser. B
Another proof is given, still using a computer, but simpler than Appel and Haken's in several respects, that every loopless planar graph admits a vertex-colouring with at most four different colours.