495 research outputs found
Spectrum of mixed bi-uniform hypergraphs
A mixed hypergraph is a triple , where is
a set of vertices, and are sets of hyperedges. A
vertex-coloring of is proper if -edges are not totally multicolored and
-edges are not monochromatic. The feasible set of is the set of
all integers, , such that has a proper coloring with colors.
Bujt\'as and Tuza [Graphs and Combinatorics 24 (2008), 1--12] gave a
characterization of feasible sets for mixed hypergraphs with all - and
-edges of the same size , .
In this note, we give a short proof of a complete characterization of all
possible feasible sets for mixed hypergraphs with all -edges of size
and all -edges of size , where . Moreover, we show that
for every sequence , , of natural numbers there
exists such a hypergraph with exactly proper colorings using colors,
, and no proper coloring with more than colors. Choosing
this answers a question of Bujt\'as and Tuza, and generalizes
their result with a shorter proof.Comment: 9 pages, 5 figure
On a problem of Henning and Yeo about the transversal number of uniform linear systems whose 2-packing number is fixed
For , let be an -uniform linear system. The
transversal number of is the minimum
number of points that intersect every line of . The 2-packing
number of is the maximum number of
lines such that the intersection of any three of them is empty. In [Discrete
Math. 313 (2013), 959--966] Henning and Yeo posed the following question: Is it
true that if is a -uniform linear system then
holds for
all ?. In this paper, some results about of -uniform linear systems
whose 2-packing number is fixed which satisfies the inequality are given
Sufficient Conditions for Tuza's Conjecture on Packing and Covering Triangles
Given a simple graph , a subset of is called a triangle cover if
it intersects each triangle of . Let and denote the
maximum number of pairwise edge-disjoint triangles in and the minimum
cardinality of a triangle cover of , respectively. Tuza conjectured in 1981
that holds for every graph . In this paper, using a
hypergraph approach, we design polynomial-time combinatorial algorithms for
finding small triangle covers. These algorithms imply new sufficient conditions
for Tuza's conjecture on covering and packing triangles. More precisely,
suppose that the set of triangles covers all edges in . We
show that a triangle cover of with cardinality at most can be
found in polynomial time if one of the following conditions is satisfied: (i)
, (ii) , (iii)
.
Keywords: Triangle cover, Triangle packing, Linear 3-uniform hypergraphs,
Combinatorial algorithm
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