53 research outputs found
Inside-Out Polytopes
We present a common generalization of counting lattice points in rational
polytopes and the enumeration of proper graph colorings, nowhere-zero flows on
graphs, magic squares and graphs, antimagic squares and graphs, compositions of
an integer whose parts are partially distinct, and generalized latin squares.
Our method is to generalize Ehrhart's theory of lattice-point counting to a
convex polytope dissected by a hyperplane arrangement. We particularly develop
the applications to graph and signed-graph coloring, compositions of an
integer, and antimagic labellings.Comment: 24 pages, 3 figures; to appear in Adv. Mat
Group Irregularity Strength of Connected Graphs
We investigate the group irregularity strength () of graphs, i.e. the
smallest value of such that taking any Abelian group \gr of order ,
there exists a function f:E(G)\rightarrow \gr such that the sums of edge
labels at every vertex are distinct. We prove that for any connected graph
of order at least 3, if and otherwise,
except the case of some infinite family of stars
Group Sum Chromatic Number of Graphs
We investigate the \textit{group sum chromatic number} (\gchi(G)) of
graphs, i.e. the smallest value such that taking any Abelian group \gr of
order , there exists a function f:E(G)\rightarrow \gr such that the sums
of edge labels properly colour the vertices. It is known that
\gchi(G)\in\{\chi(G),\chi(G)+1\} for any graph with no component of order
less than and we characterize the graphs for which \gchi(G)=\chi(G).Comment: Accepted for publication in European Journal of Combinatorics,
Elsevie
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