25 research outputs found
Cubic graphs with large circumference deficit
The circumference of a graph is the length of a longest cycle. By
exploiting our recent results on resistance of snarks, we construct infinite
classes of cyclically -, - and -edge-connected cubic graphs with
circumference ratio bounded from above by , and
, respectively. In contrast, the dominating cycle conjecture implies
that the circumference ratio of a cyclically -edge-connected cubic graph is
at least .
In addition, we construct snarks with large girth and large circumference
deficit, solving Problem 1 proposed in [J. H\"agglund and K. Markstr\"om, On
stable cycles and cycle double covers of graphs with large circumference, Disc.
Math. 312 (2012), 2540--2544]
Maximum -edge-colorable subgraphs of class II graphs
A graph is class II, if its chromatic index is at least . Let
be a maximum -edge-colorable subgraph of . The paper proves best
possible lower bounds for , and structural properties of
maximum -edge-colorable subgraphs. It is shown that every set of
vertex-disjoint cycles of a class II graph with can be extended
to a maximum -edge-colorable subgraph. Simple graphs have a maximum
-edge-colorable subgraph such that the complement is a matching.
Furthermore, a maximum -edge-colorable subgraph of a simple graph is
always class I.Comment: 13 pages, 2 figures, the proof of the Lemma 1 is correcte
Parsimonious Edge-coloring on Surfaces
We correct a small error in a 1996 paper of Albertson and Haas, and extend their lower bound for the fraction of properly colorable edges of planar subcubic graphs that are simple, connected, bridgeless, and edge-maximal to other surface embeddings of subcubic graphs
Measures of edge-uncolorability
The resistance of a graph is the minimum number of edges that have
to be removed from to obtain a graph which is -edge-colorable.
The paper relates the resistance to other parameters that measure how far is a
graph from being -edge-colorable. The first part considers regular
graphs and the relation of the resistance to structural properties in terms of
2-factors. The second part studies general (multi-) graphs . Let be
the minimum number of vertices that have to be removed from to obtain a
class 1 graph. We show that , and that this bound is best possible.Comment: 9 page