1,092 research outputs found
Exhaustive generation of -critical -free graphs
We describe an algorithm for generating all -critical -free
graphs, based on a method of Ho\`{a}ng et al. Using this algorithm, we prove
that there are only finitely many -critical -free graphs, for
both and . We also show that there are only finitely many
-critical graphs -free graphs. For each case of these cases we
also give the complete lists of critical graphs and vertex-critical graphs.
These results generalize previous work by Hell and Huang, and yield certifying
algorithms for the -colorability problem in the respective classes.
Moreover, we prove that for every , the class of 4-critical planar
-free graphs is finite. We also determine all 27 4-critical planar
-free graphs.
We also prove that every -free graph of girth at least five is
3-colorable, and determine the smallest 4-chromatic -free graph of
girth five. Moreover, we show that every -free graph of girth at least
six and every -free graph of girth at least seven is 3-colorable. This
strengthens results of Golovach et al.Comment: 17 pages, improved girth results. arXiv admin note: text overlap with
arXiv:1504.0697
Vertex colouring and forbidden subgraphs - a survey
There is a great variety of colouring concepts and results in the literature. Here our focus is to survey results on vertex colourings of graphs defined in terms of forbidden induced subgraph conditions
An Ore-type theorem for perfect packings in graphs
We say that a graph G has a perfect H-packing (also called an H-factor) if
there exists a set of disjoint copies of H in G which together cover all the
vertices of G. Given a graph H, we determine, asymptotically, the Ore-type
degree condition which ensures that a graph G has a perfect H-packing. More
precisely, let \delta_{\rm Ore} (H,n) be the smallest number k such that every
graph G whose order n is divisible by |H| and with d(x)+d(y)\geq k for all
non-adjacent x \not = y \in V(G) contains a perfect H-packing. We determine
\lim_{n\to \infty} \delta_{\rm Ore} (H,n)/n.Comment: 23 pages, 1 figure. Extra examples and a sketch proof of Theorem 4
added. To appear in the SIAM Journal on Discrete Mathematic
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