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Vertex-Coloring Edge-Weighting of Bipartite Graphs with Two Edge Weights
Let be a graph and be a subset of . A vertex-coloring
-edge-weighting of is an assignment of weight by the
elements of to each edge of so that adjacent vertices have
different sums of incident edges weights.
It was proved that every 3-connected bipartite graph admits a vertex-coloring
-edge-weighting (Lu, Yu and Zhang, (2011) \cite{LYZ}). In this paper,
we show that the following result: if a 3-edge-connected bipartite graph
with minimum degree contains a vertex such that
and is connected, then admits a vertex-coloring
-edge-weighting for . In
particular, we show that every 2-connected and 3-edge-connected bipartite graph
admits a vertex-coloring -edge-weighting for . The bound is sharp, since there exists a family of
infinite bipartite graphs which are 2-connected and do not admit
vertex-coloring -edge-weightings or vertex-coloring
-edge-weightings.Comment: In this paper, we show that every 2-connected and 3-edge-connected
bipartite graph admits a vertex-coloring S-edge-weighting for S\in
{{0,1},{1,2}
Towards the physical vacuum of cosmic inflation
There have been long debates about the initial condition of inflationary
perturbations. In this work we explicitly show the decay of excited states
during inflation via interactions. For this purpose, we note that the folded
shape non-Gaussianity can be interpreted as the decay of the non-Bunch-Davies
initial condition. The one loop diagrams with non-Bunch-Davies propagators are
calculated to uncover the decay of such excited states. The observed smallness
of non-Gaussianity keeps the window open for probing inflationary initial
conditions and trans-Planckian physics
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