2,624 research outputs found
Minimal edge colorings of class 2 graphs and double graphs
A proper edge coloring of a class 2 graph G is minimal if it contains a color class of cardinality equal to the resistance r(G) of G, which is the minimum number of edges that have to be removed from G to obtain a graph which is Δ(G)-edge colorable, where Δ(G) is the maximum degree of G. In this paper using some properties of minimal edge colorings of a class 2 graph and the notion of reflective edge colorings of the direct product of two graphs, we are able to prove that the double graph of a class 2 graph is of class 1. This result, recently conjectured, is moreover extended to some generalized double graphs
Grad and classes with bounded expansion I. decompositions
We introduce classes of graphs with bounded expansion as a generalization of
both proper minor closed classes and degree bounded classes. Such classes are
based on a new invariant, the greatest reduced average density (grad) of G with
rank r, grad r(G). For these classes we prove the existence of several
partition results such as the existence of low tree-width and low tree-depth
colorings. This generalizes and simplifies several earlier results (obtained
for minor closed classes)
Foam evaluation and Kronheimer--Mrowka theories
We introduce and study combinatorial equivariant analogues of the
Kronheimer--Mrowka homology theory of planar trivalent graphs.Comment: 53 pages, 23 tikz figure
Upper oriented chromatic number of undirected graphs and oriented colorings of product graphs
The oriented chromatic number of an oriented graph is the minimum
order of an oriented graph \vev H such that admits a homomorphism to
\vev H. The oriented chromatic number of an undirected graph is then the
greatest oriented chromatic number of its orientations. In this paper, we
introduce the new notion of the upper oriented chromatic number of an
undirected graph , defined as the minimum order of an oriented graph \vev
U such that every orientation of admits a homomorphism to . We give some properties of this parameter, derive some general upper bounds
on the ordinary and upper oriented chromatic numbers of Cartesian, strong,
direct and lexicographic products of graphs, and consider the particular case
of products of paths.Comment: 14 page
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