20,956 research outputs found
Optimal acyclic edge colouring of grid like graphs
AbstractWe determine the values of the acyclic chromatic index of a class of graphs referred to as d-dimensional partial tori. These are graphs which can be expressed as the cartesian product of d graphs each of which is an induced path or cycle. This class includes some known classes of graphs like d-dimensional meshes, hypercubes, tori, etc. Our estimates are exact except when the graph is a product of a path and a number of odd cycles, in which case the estimates differ by an additive factor of at most 1. Our results are also constructive and provide an optimal (or almost optimal) acyclic edge colouring in polynomial time
Hom complexes and homotopy theory in the category of graphs
We investigate a notion of -homotopy of graph maps that is based on
the internal hom associated to the categorical product in the category of
graphs. It is shown that graph -homotopy is characterized by the
topological properties of the \Hom complex, a functorial way to assign a
poset (and hence topological space) to a pair of graphs; \Hom complexes were
introduced by Lov\'{a}sz and further studied by Babson and Kozlov to give
topological bounds on chromatic number. Along the way, we also establish some
structural properties of \Hom complexes involving products and exponentials
of graphs, as well as a symmetry result which can be used to reprove a theorem
of Kozlov involving foldings of graphs. Graph -homotopy naturally leads
to a notion of homotopy equivalence which we show has several equivalent
characterizations. We apply the notions of -homotopy equivalence to the
class of dismantlable graphs to get a list of conditions that again
characterize these. We end with a discussion of graph homotopies arising from
other internal homs, including the construction of `-theory' associated to
the cartesian product in the category of reflexive graphs.Comment: 28 pages, 13 figures, final version, to be published in European J.
Com
Strong geodetic problem on Cartesian products of graphs
The strong geodetic problem is a recent variation of the geodetic problem.
For a graph , its strong geodetic number is the cardinality of
a smallest vertex subset , such that each vertex of lies on a fixed
shortest path between a pair of vertices from . In this paper, the strong
geodetic problem is studied on the Cartesian product of graphs. A general upper
bound for is determined, as well as exact values
for , , and certain prisms.
Connections between the strong geodetic number of a graph and its subgraphs are
also discussed.Comment: 18 pages, 9 figure
Hypercellular graphs: partial cubes without as partial cube minor
We investigate the structure of isometric subgraphs of hypercubes (i.e.,
partial cubes) which do not contain finite convex subgraphs contractible to the
3-cube minus one vertex (here contraction means contracting the edges
corresponding to the same coordinate of the hypercube). Extending similar
results for median and cellular graphs, we show that the convex hull of an
isometric cycle of such a graph is gated and isomorphic to the Cartesian
product of edges and even cycles. Furthermore, we show that our graphs are
exactly the class of partial cubes in which any finite convex subgraph can be
obtained from the Cartesian products of edges and even cycles via successive
gated amalgams. This decomposition result enables us to establish a variety of
results. In particular, it yields that our class of graphs generalizes median
and cellular graphs, which motivates naming our graphs hypercellular.
Furthermore, we show that hypercellular graphs are tope graphs of zonotopal
complexes of oriented matroids. Finally, we characterize hypercellular graphs
as being median-cell -- a property naturally generalizing the notion of median
graphs.Comment: 35 pages, 6 figures, added example answering Question 1 from earlier
draft (Figure 6.
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