2 research outputs found
On a property of the -dimensional cube
We show that in any subset of the vertices of -dimensional cube that
contains at least vertices (), there are four vertices
that induce a claw, or there are eight vertices that induce the cycle of length
eight.Comment: 2 pages, no figure
On the extremal values of the number of vertices with an interval spectrum on the set of proper edge colorings of the graph of the -dimensional cube
For an undirected, simple, finite, connected graph , we denote by
and the sets of its vertices and edges, respectively. A function
is called a proper edge -coloring of a
graph , if adjacent edges are colored differently and each of colors is
used. The least value of for which there exists a proper edge -coloring
of a graph is denoted by . For any graph , and for any integer
satisfying the inequality , we denote by
the set of all proper edge -colorings of . Let us also
define a set of all proper edge colorings of a graph :
An arbitrary nonempty finite subset of consecutive integers is called an
interval. If and , then the set of colors of
edges of which are incident with is denoted by and is
called a spectrum of the vertex of the graph at the proper edge
coloring . If is a graph and , then define
.
For a graph and any integer , satisfying the inequality , we define:
For any graph , we set:
For any positive integer , the exact values of the parameters ,
, and are found for the graph of the
-dimensional cube.Comment: 9 pages. arXiv admin note: substantial text overlap with
arXiv:1205.012