Li, Nikiforov and Schelp [13] conjectured that any 2-edge coloured graph G with order n and minimum degree δ(G) > 3n/4 contains a monochromatic cycle of length ℓ, for all ℓ ∈ [4, ⌈n/2⌉]. We prove this conjecture for sufficiently large n and also find all 2-edge coloured graphs with δ(G)=3n/4 that do not contain all such cycles. Finally, we show that, for all δ>0 and n>n0(δ), if G is a 2-edge coloured graph of order n with δ(G) ≥ 3n/4, then one colour class either contains a monochromatic cycle of length at least (2/3+δ/2)n, or contains monochromatic cycles of all lengths ℓ ∈ [3, (2/3−δ)n].
Bootstrap percolation, one of the simplest cellular automata, can be seen as a model of the spread of infection. In $r$-neighbour bootstrap percolation on a graph $G$ we assign a state, infected or healthy, to every vertex of $G$ and then update these states in successive rounds, according to the following simple local update rule: infected vertices of $G$ remain infected forever and a healthy vertex becomes infected if it has at least $r$ already infected neighbours. We say that percolation occurs if eventually every vertex of $G$ becomes infected. A well known and celebrated fact about the classical model of $2$-neighbour bootstrap percolation on the $n \times n$ square grid is that the smallest size of an initially infected set which percolates in this process is $n$. In this paper we consider the problem of finding the maximum time a $2$-neighbour bootstrap process on $[n]^2$ with $n$ initially infected vertices can take to eventually infect the entire vertex set. Answering a question posed by Bollobás we compute the exact value for this maximum showing that, for $n \ge 4$, it is equal to the integer nearest to $(5n^2-2n)/8$.
We consider a classic model known as bootstrap percolation on the n × n square grid. To each vertex of the grid we assign an initial state, infected or healthy, and then in consecutive rounds we infect every healthy vertex that has at least two already infected neighbors. We say that percolation occurs if the whole grid is eventually infected. In this paper, contributing to a recent series of extremal results in this field, we prove that the maximum time a bootstrap percolation process can take to eventually infect the entire vertex set of the grid is 13n 2 /18 + O(n).
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