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Register a new userOre and Chvatal-type Degree Conditions for Bootstrap Percolation from Small Sets
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Bootstrap percolation is a deterministic cellular automaton in which vertices of a graph~$G$ begin in one of two states, dormant or active. Given a fixed integer $r$, a dormant vertex becomes active if at any stage it has at least $r$ active neighbors, and it remains active for the duration of the process. Given an initial set of active vertices $A$, we say that $G$ $r$-percolates (from $A$) if every vertex in $G$ becomes active after some number of steps. Let $m(G,r)$ denote the minimum size of a set $A$ such that $G$ $r$-percolates from $A$. Bootstrap percolation has been studied in a number of settings, and has applications to both statistical physics and discrete epidemiology. Here, we are concerned with degree-based density conditions that ensure $m(G,2)=2$. In particular, we give an Ore-type degree sum result that states that if a graph $G$ satisfies $sigma_2(G)ge n-2$, then either $m(G,2)=2$ or $G$ is in one of a small number of classes of exceptional graphs. We also give a Chv{a}tal-type degree condition: If $G$ is a graph with degree sequence $d_1le d_2ledotsle d_n$ such that $d_i geq i+1$ or $d_{n-i} geq n-i-1$ for all $1 leq i < frac{n}{2}$, then $m(G,2)=2$ or $G$ falls into one of several specific exceptional classes of graphs. Both of these results are inspired by, and extend, an Ore-type result in [D. Freund, M. Poloczek, and D. Reichman, Contagious sets in dense graphs, to appear in European J. Combin.]
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By bootstrap percolation we mean the following deterministic process on a graph $G$. Given a set $A$ of vertices infected at time 0, new vertices are subsequently infected, at each time step, if they have at least $rinmathbb{N}$ previously infected neighbors. When the set $A$ is chosen at random, the main aim is to determine the critical probability $p_c(G,r)$ at which percolation (infection of the entire graph) becomes likely to occur. This bootstrap process has been extensively studied on the $d$-dimensional grid $[n]^d$: with $2leq rleq d$ fixed, it was proved by Cerf and Cirillo (for $d=r=3$), and by Cerf and Manzo (in general), that [p_c([n]^d,r)=Thetabiggl(frac{1}{log_{(r-1)}n}biggr)^{d-r+1},] where $log_{(r)}$ is an $r$-times iterated logarithm. However, the exact threshold function is only known in the case $d=r=2$, where it was shown by Holroyd to be $(1+o(1))frac{pi^2}{18log n}$. In this paper we shall determine the exact threshold in the crucial case $d=r=3$, and lay the groundwork for solving the problem for all fixed $d$ and $r$.
Caccetta-Haggkvist conjecture is a longstanding open problem on degree conditions that force an oriented graph to contain a directed cycle of a bounded length. Motivated by this conjecture, Kelly, Kuhn and Osthus initiated a study of degree conditions forcing the containment of a directed cycle of a given length. In particular, they found the optimal minimum semidegree, i.e., the smaller of the minimum indegree and the minimum outdegree, that forces a large oriented graph to contain a directed cycle of a given length not divisible by $3$, and conjectured the optimal minimum semidegree for all the other cycles except the directed triangle. In this paper, we establish the best possible minimum semidegree that forces a large oriented graph to contain a directed cycle of a given length divisible by $3$ yet not equal to $3$, hence fully resolve the conjecture of Kelly, Kuhn and Osthus. We also find an asymptotically optimal semidegree threshold of any cycle with a given orientation of its edges with the sole exception of a directed triangle.
Let (G, +) be an abelian group. A subset of G is sumfree if it contains no elements x, y, z such that x +y = z. We extend this concept by introducing the Schur degree of a subset of G, where Schur degree 1 corresponds to sumfree. The classical inequality S(n) $le$ R n (3) -- 2, between the Schur number S(n) and the Ramsey number R n (3) = R(3,. .. , 3), is shown to remain valid in a wider context, involving the Schur degree of certain subsets of G. Recursive upper bounds are known for R n (3) but not for S(n) so far. We formulate a conjecture which, if true, would fill this gap. Indeed, our study of the Schur degree leads us to conjecture S(n) $le$ n(S(n -- 1) + 1) for all n $ge$ 2. If true, it would yield substantially better upper bounds on the Schur numbers, e.g. S(6) $le$ 966 conjecturally, whereas all is known so far is 536 $le$ S(6) $le$ 1836.
Desingularization is the problem of finding a left multiple of a given Ore operator in which some factor of the leading coefficient of the original operator is removed. An order-degree curve for a given Ore operator is a curve in the $(r,d)$-plane such that for all points $(r,d)$ above this curve, there exists a left multiple of order $r$ and degree $d$ of the given operator. We give a new proof of a desingularization result by Abramov and van Hoeij for the shift case, and show how desingularization implies order-degree curves which are extremely accurate in examples.
Given a simple graph $G$, denote by $Delta(G)$, $delta(G)$, and $chi(G)$ the maximum degree, the minimum degree, and the chromatic index of $G$, respectively. We say $G$ is emph{$Delta$-critical} if $chi(G)=Delta(G)+1$ and $chi(H)le Delta(G)$ for every proper subgraph $H$ of $G$; and $G$ is emph{overfull} if $|E(G)|>Delta lfloor |V(G)|/2 rfloor$. Since a maximum matching in $G$ can have size at most $lfloor |V(G)|/2 rfloor$, it follows that $chi(G) = Delta(G) +1$ if $G$ is overfull. Conversely, let $G$ be a $Delta$-critical graph. The well known overfull conjecture of Chetwynd and Hilton asserts that $G$ is overfull provided $Delta(G) > |V(G)|/3$. In this paper, we show that any $Delta$-critical graph $G$ is overfull if $Delta(G) - 7delta(G)/4ge(3|V(G)|-17)/4$.
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