30,206 research outputs found
On congruence equations arising from suborbital graphs
In this paper we deal with congruence equations arising from suborbital graphs of the normalizer of Γ_0(m) in PSL(2,R) . We also propose a conjecture concerning the suborbital graphs of the normalizer and the related congruence equations. In order to prove the existence of solution of an equation over prime finite field, this paper utilizes the Fuchsian group action on the upper half plane and Farey graphs properties
Minimum Number of k-Cliques in Graphs with Bounded Independence Number
Erdos asked in 1962 about the value of f(n,k,l), the minimum number of
k-cliques in a graph of order n and independence number less than l. The case
(k,l)=(3,3) was solved by Lorden. Here we solve the problem (for all large n)
when (k,l) is (3,4), (3,5), (3,6), (3,7), (4,3), (5,3), (6,3), and (7,3).
Independently, Das, Huang, Ma, Naves, and Sudakov did the cases (k,l)=(3,4) and
(4,3).Comment: 25 pages. v4: Three new solved cases added: (3,5), (3,6), (3,7). All
calculations are done with Version 2.0 of Flagmatic no
Non-singular circulant graphs and digraphs
We give necessary and sufficient conditions for a few classes of known
circulant graphs and/or digraphs to be singular. The above graph classes are
generalized to -digraphs for non-negative integers and , and
the digraph , with certain restrictions. We also obtain a
necessary and sufficient condition for the digraphs to be
singular. Some necessary conditions are given under which the
-digraphs are singular.Comment: 12 page
About the existence of oriented paths with three blocks
A path P(k,l,r) is an oriented path consisting of k forward arcs, followed by
l backward arcs, and then by r forward arcs. We prove the existence of any
oriented path of length n-1 with three blocks having the middle block of length
one in any (2n-3)- chromatic digraph, which is an improvement of the latest
bound reached in this case. Concerning the general case of paths with three
blocks, we prove, after partitioning the problem into three cases according to
the value of k,l and r that the chromatic number of digraphs containing no
P(k,l,r) of length n-1 is bounded above by 2(n-1)+r, 2(n-1)+l+r-k and
2(n+l-1)-k in the three cases respectively
On decision and optimization (k,l)-graph sandwich problems
AbstractA graph G is (k,l) if its vertex set can be partitioned into at most k independent sets and l cliques. The (k,l)-Graph Sandwich Problem asks, given two graphs G1=(V,E1) and G2=(V,E2), whether there exists a graph G=(V,E) such that E1⊆E⊆E2 and G is (k,l). In this paper, we prove that the (k,l)-Graph Sandwich Problem is NP-complete for the cases k=1 and l=2; k=2 and l=1; or k=l=2. This completely classifies the complexity of the (k,l)-Graph Sandwich Problem as follows: the problem is NP-complete, if k+l>2; the problem is polynomial otherwise. We consider the degree Δ constraint subproblem and completely classify the problem as follows: the problem is polynomial, for k⩽2 or Δ⩽3; the problem is NP-complete otherwise. In addition, we propose two optimization versions of graph sandwich problem for a property Π: MAX-Π-GSP and MIN-Π-GSP. We prove that MIN-(2,1)-GSP is a Max-SNP-hard problem, i.e., there is a positive constant ε, such that the existence of an ε-approximative algorithm for MIN-(2,1)-GSP implies P=NP
On Polygons Excluding Point Sets
By a polygonization of a finite point set in the plane we understand a
simple polygon having as the set of its vertices. Let and be sets
of blue and red points, respectively, in the plane such that is in
general position, and the convex hull of contains interior blue points
and interior red points. Hurtado et al. found sufficient conditions for the
existence of a blue polygonization that encloses all red points. We consider
the dual question of the existence of a blue polygonization that excludes all
red points . We show that there is a minimal number , which is
polynomial in , such that one can always find a blue polygonization
excluding all red points, whenever . Some other related problems are
also considered.Comment: 14 pages, 15 figure
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