4,287 research outputs found
The independence number in graphs of maximum degree three
We prove that a K4-free graph G of order n, size m and maximum degree at most three has an independent set of cardinality at least 1/7 (4n − m − \lambda − tr) where \lambda counts the number of components of G whose blocks are each either isomorphic to one of four specific graphs or edges between two of these four specific graphs and tr is the maximum number of vertex-disjoint triangles in G. Our result generalizes a bound due to Heckman and Thomas (A New Proof of the Independence Ratio of Triangle-Free Cubic Graphs, Discrete Math. 233 (2001), 233-237)
Triangles in graphs without bipartite suspensions
Given graphs and , the generalized Tur\'an number ex is the
maximum number of copies of in an -vertex graph with no copies of .
Alon and Shikhelman, using a result of Erd\H os, determined the asymptotics of
ex when the chromatic number of is greater than 3 and proved
several results when is bipartite. We consider this problem when has
chromatic number 3. Even this special case for the following relatively simple
3-chromatic graphs appears to be challenging.
The suspension of a graph is the graph obtained from by
adding a new vertex adjacent to all vertices of . We give new upper and
lower bounds on ex when is a path, even cycle, or
complete bipartite graph. One of the main tools we use is the triangle removal
lemma, but it is unclear if much stronger statements can be proved without
using the removal lemma.Comment: New result about path with 5 edges adde
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
On two problems in Ramsey-Tur\'an theory
Alon, Balogh, Keevash and Sudakov proved that the -partite Tur\'an
graph maximizes the number of distinct -edge-colorings with no monochromatic
for all fixed and , among all -vertex graphs. In this
paper, we determine this function asymptotically for among -vertex
graphs with sub-linear independence number. Somewhat surprisingly, unlike
Alon-Balogh-Keevash-Sudakov's result, the extremal construction from
Ramsey-Tur\'an theory, as a natural candidate, does not maximize the number of
distinct edge-colorings with no monochromatic cliques among all graphs with
sub-linear independence number, even in the 2-colored case.
In the second problem, we determine the maximum number of triangles
asymptotically in an -vertex -free graph with . The
extremal graphs have similar structure to the extremal graphs for the classical
Ramsey-Tur\'an problem, i.e.~when the number of edges is maximized.Comment: 22 page
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