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Quasirandom Latin squares
We prove a conjecture by Garbe et al. [arXiv:2010.07854] by showing that a Latin square is quasirandom if and only if the density of every 2x3 pattern is 1/720+o(1). This result is the best possible
Long cycles and spanning subgraphs of locally maximal 1‐planar graphs
TLDR
This work shows the existence of a spanning $3-connected planar subgraph and proves that G is hamiltonian if $G$ has at most three $3$-vertex-cuts, and that the graph is traceable if G has at least four four $3#-ver Tex-cuts.
Random Perturbation of Sparse Graphs
TLDR
This note extends the result by Bohman, Frieze, and Martin on the threshold in $\mathbb{G}(n,p)$ to sparser graphs with $\alpha=o(1)$, and discusses embeddings of bounded degree trees and other spanning structures in this model.
Longer Cycles in Essentially 4-Connected Planar Graphs
TLDR
It is proved that an essentially 4-connected planar graph on n vertices contains a cycle of length at least 35(n+2) {3 \over 5} ( {n + 2} ) and that such a cycle can be found in time O(n2).
On Selkow’s Bound on the Independence Number of Graphs
TLDR
A new probabilistic proof is given of Selkow's bound on α (G), where N(v) and d( v) = |N(v)| denote the neighborhood and the degree of a vertex v ∈ V (G).
On the Circumference of Essentially 4-connected Planar Graphs
TLDR
It is proved that every essentially 4-connected planar graph G on n vertices contains a cycle of length at least $\frac{5}{8}(n+2)$.
Cycles through a set of specified vertices of a planar graph
. Confirming a conjecture of Plummer, Thomas and Yu proved that a 4-connected planar graph contains a cycle through all but two (freely choosable) vertices. Here we prove that a planar graph G
Circumference of essentially 4-connected planar triangulations
TLDR
It is proved that every essentially 4-connected maximal planar graph G on n vertices contains a cycle of length at least 2 3 (n+ 4); moreover, this bound is sharp.
Kempe Chains and Rooted Minors
A (minimal) transversal of a partition is a set which contains exactly one element from each member of the partition and nothing else. A coloring of a graph is a partition of its vertex set into
Uniform Tur\'an density of cycles
In the early 1980s, Erdős and Sós initiated the study of the classical Turán problem with a uniformity condition: the uniform Turán density of a hypergraph H is the infimum over all d for which any
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