391 research outputs found
Convex circuit free coloration of an oriented graph
We introduce the \textit{convex circuit-free coloration} and \textit{convex circuit-free chromatic number} of an oriented graph and establish various basic results. We show that the problem of deciding if an oriented graph verifies is NP-complete if and polynomial if . We exhibit an algorithm which finds the optimal convex circuit-free coloration for tournaments, and characterize the tournaments that are \textit{vertex-critical} for the convex circuit-free coloration
Lattice Points in Orthotopes and a Huge Polynomial Tutte Invariant of Weighted Gain Graphs
A gain graph is a graph whose edges are orientably labelled from a group. A
weighted gain graph is a gain graph with vertex weights from an abelian
semigroup, where the gain group is lattice ordered and acts on the weight
semigroup. For weighted gain graphs we establish basic properties and we
present general dichromatic and forest-expansion polynomials that are Tutte
invariants (they satisfy Tutte's deletion-contraction and multiplicative
identities). Our dichromatic polynomial includes the classical graph one by
Tutte, Zaslavsky's two for gain graphs, Noble and Welsh's for graphs with
positive integer weights, and that of rooted integral gain graphs by Forge and
Zaslavsky. It is not a universal Tutte invariant of weighted gain graphs; that
remains to be found.
An evaluation of one example of our polynomial counts proper list colorations
of the gain graph from a color set with a gain-group action. When the gain
group is Z^d, the lists are order ideals in the integer lattice Z^d, and there
are specified upper bounds on the colors, then there is a formula for the
number of bounded proper colorations that is a piecewise polynomial function of
the upper bounds, of degree nd where n is the order of the graph.
This example leads to graph-theoretical formulas for the number of integer
lattice points in an orthotope but outside a finite number of affinographic
hyperplanes, and for the number of n x d integral matrices that lie between two
specified matrices but not in any of certain subspaces defined by simple row
equations.Comment: 32 pp. Submitted in 2007, extensive revisions in 2013 (!). V3: Added
references, clarified examples. 35 p
An extensive English language bibliography on graph theory and its applications, supplement 1
Graph theory and its applications - bibliography, supplement
Inversion dans les tournois
We consider the transformation reversing all arcs of a subset of the
vertex set of a tournament . The \emph{index} of , denoted by , is
the smallest number of subsets that must be reversed to make acyclic. It
turns out that critical tournaments and -critical tournaments can be
defined in terms of inversions (at most two for the former, at most four for
the latter). We interpret as the minimum distance of to the
transitive tournaments on the same vertex set, and we interpret the distance
between two tournaments and as the \emph{Boolean dimension} of a
graph, namely the Boolean sum of and . On vertices, the maximum
distance is at most , whereas , the maximum of over the
tournaments on vertices, satisfies , for . Let (resp.
) be the class of finite (resp. at most
countable) tournaments such that . The class is determined by finitely many obstructions. We give a
morphological description of the members of and a
description of the critical obstructions. We give an explicit description of an
universal tournament of the class .Comment: 6 page
Spectral behavior of some graph and digraph compositions
Let G be a graph of order n the vertices of which are labeled from 1 to n and let , · · · , be n graphs. The graph composition G[, · · · ,] is the graph obtained by replacing the vertex i of G by the graph Gi and there is an edge between u ∈ and v ∈ if and only if there is an edge between i and j in G. We first consider graph composition G[, · · · ,] where G is regular and is a complete graph and we establish
some links between the spectral characterisation of G and the spectral characterisation of G[, · · · ,]. We then prove that two non isomorphic graphs G[, · · ·] where are complete
graphs and G is a strict threshold graph or a star are not Laplacian-cospectral, giving rise to a spectral characterization
of these graphs. We also consider directed graphs, especially the vertex-critical tournaments without non-trivial acyclic interval which are tournaments of the shape t[, · · · ,], where t
is a tournament and is a circulant tournament. We give
conditions to characterise these graphs by their spectrum.Peer Reviewe
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