7,509 research outputs found
On pseudo 2-factors
AbstractWe show that a graph with minimum degree δ, independence number α≥δ and without isolated vertices, possesses a partition by vertex-disjoint cycles and at most α−δ+1 edges or vertices
Independence Number and Disjoint Theta Graphs
The goal of this paper is to find vertex disjoint even cycles in graphs. For this purpose, define a θ-graph to be a pair of vertices u,v with three internally disjoint paths joining u to v. Given an independence number α and a fixed integer k, the results contained in this paper provide sharp bounds on the order f(k,α) of a graph with independence number α(G)≤α which contains no k disjoint θ-graphs. Since every θ-graph contains an even cycle, these results provide k disjoint even cycles in graphs of order at least f(k,α)+1. We also discuss the relationship between this problem and a generalized ramsey problem involving sets of graphs
Partitioning 2-edge-colored graphs by monochromatic paths and cycles
We present results on partitioning the vertices of -edge-colored graphs
into monochromatic paths and cycles. We prove asymptotically the two-color case
of a conjecture of S\'ark\"ozy: the vertex set of every -edge-colored graph
can be partitioned into at most monochromatic cycles, where
denotes the independence number of . Another direction, emerged
recently from a conjecture of Schelp, is to consider colorings of graphs with
given minimum degree. We prove that apart from vertices, the vertex
set of any -edge-colored graph with minimum degree at least
(1+\eps){3|V(G)|\over 4} can be covered by the vertices of two vertex
disjoint monochromatic cycles of distinct colors. Finally, under the assumption
that does not contain a fixed bipartite graph , we show that
in every -edge-coloring of , vertices can be covered by two
vertex disjoint paths of different colors, where is a constant depending
only on . In particular, we prove that , which is best possible
The Erdős-Ko-Rado properties of various graphs containing singletons
Let G=(V,E) be a graph. For r≥1, let be the family of independent vertex r-sets of G. For vV(G), let denote the star . G is said to be r-EKR if there exists vV(G) such that for any non-star family of pair-wise intersecting sets in . If the inequality is strict, then G is strictly r-EKR.
Let Γ be the family of graphs that are disjoint unions of complete graphs, paths, cycles, including at least one singleton. Holroyd, Spencer and Talbot proved that, if GΓ and 2r is no larger than the number of connected components of G, then G is r-EKR. However, Holroyd and Talbot conjectured that, if G is any graph and 2r is no larger than μ(G), the size of a smallest maximal independent vertex set of G, then G is r-EKR, and strictly so if 2r<μ(G). We show that in fact, if GΓ and 2r is no larger than the independence number of G, then G is r-EKR; we do this by proving the result for all graphs that are in a suitable larger set Γ′Γ. We also confirm the conjecture for graphs in an even larger set Γ″Γ′
3-uniform hypergraphs and linear cycles
We continue the work of Gyárfás, Győri and Simonovits [Gyárfás, A., E. Győri and M. Simonovits, On 3-uniform hypergraphs without linear cycles. Journal of Combinatorics 7 (2016), 205–216], who proved that if a 3-uniform hypergraph H with n vertices has no linear cycles, then its independence number α≥[Formula presented]. The hypergraph consisting of vertex disjoint copies of complete hypergraphs K5 3 shows that equality can hold. They asked whether α can be improved if we exclude K5 3 as a subhypergraph and whether such a hypergraph is 2-colorable. We answer these questions affirmatively. Namely, we prove that if a 3-uniform linear-cycle-free hypergraph H, doesn't contain K5 3 as a subhypergraph, then it is 2-colorable. This result clearly implies that α≥⌈[Formula presented]⌉. We show that this bound is sharp. Gyárfás, Győri and Simonovits also proved that a linear-cycle-free 3-uniform hypergraph contains a vertex of strong degree at most 2. In this context, we show that a linear-cycle-free 3-uniform hypergraph has a vertex of degree at most n−2 when n≥10. © 2017 Elsevier B.V
Monochromatic cycle covers in random graphs
A classic result of Erd\H{o}s, Gy\'arf\'as and Pyber states that for every
coloring of the edges of with colors, there is a cover of its vertex
set by at most vertex-disjoint monochromatic cycles. In
particular, the minimum number of such covering cycles does not depend on the
size of but only on the number of colors. We initiate the study of this
phenomena in the case where is replaced by the random graph . Given a fixed integer and , we
show that with high probability the random graph has
the property that for every -coloring of the edges of , there is a
collection of monochromatic cycles covering all the
vertices of . Our bound on is close to optimal in the following sense:
if , then with high probability there are colorings of
such that the number of monochromatic cycles needed to
cover all vertices of grows with .Comment: 24 pages, 1 figure (minor changes, added figure
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