179 research outputs found
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}
Sigma Partitioning: Complexity and Random Graphs
A of a graph is a partition of the vertices
into sets such that for every two adjacent vertices and
there is an index such that and have different numbers of
neighbors in . The of a graph , denoted by
, is the minimum number such that has a sigma partitioning
. Also, a of a graph is a
function , such that for every two adjacent
vertices and of , ( means that and are adjacent). The of , denoted by , is the minimum number such
that has a lucky labeling . It was
conjectured in [Inform. Process. Lett., 112(4):109--112, 2012] that it is -complete to decide whether for a given 3-regular
graph . In this work, we prove this conjecture. Among other results, we give
an upper bound of five for the sigma number of a uniformly random graph
A Solution to the 1-2-3 Conjecture
We show that for every graph without isolated edge, the edges can be assigned
weights from {1,2,3} so that no two neighbors receive the same sum of incident
edge weights. This solves a conjecture of Karo\'{n}ski, Luczak, and Thomason
from 2004.Comment: 16 page
A determinant formula for the Jones polynomial of pretzel knots
This paper presents an algorithm to construct a weighted adjacency matrix of
a plane bipartite graph obtained from a pretzel knot diagram. The determinant
of this matrix after evaluation is shown to be the Jones polynomial of the
pretzel knot by way of perfect matchings (or dimers) of this graph. The weights
are Tutte's activity letters that arise because the Jones polynomial is a
specialization of the signed version of the Tutte polynomial. The relationship
is formalized between the familiar spanning tree setting for the Tait graph and
the perfect matchings of the plane bipartite graph above. Evaluations of these
activity words are related to the chain complex for the Champanerkar-Kofman
spanning tree model of reduced Khovanov homology.Comment: 19 pages, 12 figures, 2 table
d-Lucky Labeling of Graphs
AbstractLet l: V (G) →N be a labeling of the vertices of a graph G by positive integers. Define , where d(u) denotes the degree of u and N(u) denotes the open neighborhood of u. In this paper we introduce a new labeling called d-lucky labeling and study the same as a vertex coloring problem. We define a labeling l as d-lucky if c(u) ≠c(v) , for every pair of adjacent vertices u and v in G. The d-lucky number of a graph G, denoted by ηdl(G), is the least positive k such that G has a d-lucky labeling with {1,2, ..., k} as the set of labels. We obtain ηdl(G) = 2 for hypercube network, butterfly network, benes network, mesh network, hypertree and X-tree
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