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Abstract:
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We present original results from the following fields of discrete
mathematics: chromatic graph theory, extremal set theory and Boolean matrix
theory.
From the chromatic graph theory we investigate edge and total colorings
satisfying the condition that neighboring vertices of a graph possess different
values of multi-set, set or sum, induced by the giving coloring.
Multi-set neighbor-distinguishing edge coloring of a graph is an assignment
of colors to edges such that, for every edge uv of a graph, multi-set of
the edges incident with the vertex u differs from the multi-set of the edges
incident with the vertex v. The previous best result concerning the minimum
number of colors required for such a coloring of an arbitrary graph states that
four colors are sufficient. The author’s contribution is a proof that such a
coloring is always possible with only three colors, which is in general case the
optimal number of colors.
We construct a graph for which we subsequently prove that a different
number of colors is required to obtain a multi-set neighbor-distinguishing
coloring and neighbor-distinguishing coloring by sum. As far as we know,
this is the first example of such a graph.
A few results concerning the neighbor expended sum distinguishing coloring
are given. The main contribution is a proof that for an arbitrary graph
there exists a total coloring from the set f1; 2; 3g, such that every two adjacent
vertices have different sums of its adjacent vertices and incident edges.
Also, for certain classes of graphs is proved that there exists such a coloring
using only the colors from the set f1; 2g.
Neighbor-distinguishing edge coloring of a graph G requires that every two
adjacent edges receive different colors, while the sets of the edges incident
with the vertices u and v differ for every edge uv of G. The author presents
a procedure of edge coloring for an arbitrary graph without isolated edges,
where we a smaller number of colors is used compared to all known results.
For the adjacent vertex distinguishing total coloring of a graph G the
condition is that every two adjacent and incident elements of V (G) [ E(G)
receive different colors, while for every edge uv of G the set composed from
the colors assigned to the edges incident with u together with the color of
u, differs from such a set for v. The author improves the upper bound of
the minimum number of colors needed for such a coloring, relative to the
maximal degree of a graph.
Frankl’s conjecture from the extremal set theory states that for every
family closed under union there exists an element contained in at least half
of the sets of the family.
We give a proof that Frankl’s conjecture holds for every family contained
from 12 elements, while it is known that this is true for families contained
from 11 or less elements. Our proof is based on the efficient algorithm that
exhausts all the possibilities, while using the results for subfamilies that
eventual counter-example cannot contain, which we obtained in a number of
consecutive steps.
Family of sets G is an FC-family if for every family F containing G there
exists an element from
S
G that appears in at least half of the sets of F.
NonFC-family is every family that is not FC. The author’s contribution is
the complete classification of all families consisting of 6 or less elements into
FC and NonFC-families.
From the Boolean matrices theory we present our results concerning the
row space cardinality. Boolean matrices are the matrices whose all components
are from the set f0; 1g, while the row space of a Boolean matrix is the
set of vectors that can be obtained by disjunction from the rows of a matrix.
We present the set consisted of all values a from the interval [2n2 +
2n3; 2n2] such that there exists a matrix of dimension n n having the row
space cardinality equal to a.
For the least positive integer an for which there exists no matrix of dimension
n n having the row space cardinality equal to an, the author gives
a lower bound that is an improvement over the previously known results.
All proofs for the main results in the dissertation are constructive. Proofs
of some of them require the use of computers where there is a calculation of a
great number of possibilities. For other proofs this was not necessity, though
algorithms following the steps of the proofs can be implemented to obtain a
graph coloring or a matrix with the desired properties. |