5,776 research outputs found
Even-cycle decompositions of graphs with no odd--minor
An even-cycle decomposition of a graph G is a partition of E(G) into cycles
of even length. Evidently, every Eulerian bipartite graph has an even-cycle
decomposition. Seymour (1981) proved that every 2-connected loopless Eulerian
planar graph with an even number of edges also admits an even-cycle
decomposition. Later, Zhang (1994) generalized this to graphs with no
-minor.
Our main theorem gives sufficient conditions for the existence of even-cycle
decompositions of graphs in the absence of odd minors. Namely, we prove that
every 2-connected loopless Eulerian odd--minor-free graph with an even
number of edges has an even-cycle decomposition.
This is best possible in the sense that `odd--minor-free' cannot be
replaced with `odd--minor-free.' The main technical ingredient is a
structural characterization of the class of odd--minor-free graphs, which
is due to Lov\'asz, Seymour, Schrijver, and Truemper.Comment: 17 pages, 6 figures; minor revisio
A Study on Integer Additive Set-Valuations of Signed Graphs
Let denote the set of all non-negative integers and \cP(\N) be its
power set. An integer additive set-labeling (IASL) of a graph is an
injective set-valued function f:V(G)\to \cP(\N)-\{\emptyset\} such that the
induced function f^+:E(G) \to \cP(\N)-\{\emptyset\} is defined by , where is the sumset of and . A graph
which admits an IASL is usually called an IASL-graph. An IASL of a graph
is said to be an integer additive set-indexer (IASI) of if the
associated function is also injective. In this paper, we define the
notion of integer additive set-labeling of signed graphs and discuss certain
properties of signed graphs which admits certain types of integer additive
set-labelings.Comment: 12 pages, Carpathian Mathematical Publications, Vol. 8, Issue 2,
2015, 12 page
P-matrices and signed digraphs
We associate a signed digraph with a list of matrices whose dimensions permit
them to be multiplied, and whose product is square. Cycles in this graph have a
parity, that is, they are either even (termed e-cycles) or odd (termed
o-cycles). The absence of e-cycles in the graph is shown to imply that the
matrix product is a P0-matrix, i.e., all of its principal minors are
nonnegative. Conversely, the presence of an e-cycle is shown to imply that
there exists a list of matrices associated with the graph whose product fails
to be a P0-matrix. The results generalise a number of previous results relating
P- and P0-matrices to graphs
Graph-theoretic approaches to injectivity and multiple equilibria in systems of interacting elements
We extend previous work on injectivity in chemical reaction networks to
general interaction networks. Matrix- and graph-theoretic conditions for
injectivity of these systems are presented. A particular signed, directed,
labelled, bipartite multigraph, termed the ``DSR graph'', is shown to be a
useful representation of an interaction network when discussing questions of
injectivity. A graph-theoretic condition, developed previously in the context
of chemical reaction networks, is shown to be sufficient to guarantee
injectivity for a large class of systems. The graph-theoretic condition is
simple to state and often easy to check. Examples are presented to illustrate
the wide applicability of the theory developed.Comment: 34 pages, minor corrections and clarifications on previous versio
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