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We consider a variation on Maker-Breaker games on graphs or digraphs where the edges have random costs. We assume that Maker wishes to choose the edges of a spanning tree, but wishes to minimise his cost. Meanwhile Breaker wants to make Maker's cost as large as possible. 

ABSTRACT We discuss the existence of Hamilton cycles in the random graph where there are restrictions caused by (i) coloring sequences, (ii) a subset of vertices must occur in a specific order, and (iii) there is a bound on the number of inversions in the associated permutation. 

ABSTRACT We study the fixation probability for two versions of the Moran process on the random graph at the threshold for connectivity. The Moran process models the spread of a mutant population in a network. Throughout the process, there are vertices of two types, mutants, and non‐mutants. Mutants have fitness and non‐mutants have fitness 1. The process starts with a unique individual mutant located at the vertex . In the Birth‐Death version of the process a random vertex is chosen proportionally to its fitness and then changes the type of a random neighbor to its own. The process continues until the set of mutants is empty or . In the Death‐Birth version, a uniform random vertex is chosen and then takes the type of a random neighbor, chosen according to fitness. The process again continues until the set of mutants is empty or . Thefixation probabilityis the probability that the process ends with . We show that asymptotically correct estimates of the fixation probability depend only on the degree of and its neighbors. In some cases we can provide values for these estimates and in other places we can only provide non‐linear recurrences that could be used to compute values. 

The greedy and nearest-neighbor TSP heuristics can both have $$\log n$$ approximation factors from optimal in worst case, even just for $$n$$ points in Euclidean space. In this note, we show that this approximation factor is only realized when the optimal tour is unusually short. In particular, for points from any fixed $$d$$-Ahlfor's regular metric space (which includes any $$d$$-manifold like the $$d$$-cube $[0,1]^d$ in the case $$d$$ is an integer but also fractals of dimension $$d$$ when $$d$$ is real-valued), our results imply that the greedy and nearest-neighbor heuristics have additive errors from optimal on the order of the optimal tour length through random points in the same space, for $d>1$. 

Let $$N=\binom{n}{2}$$ and $$s\geq 2$$. Let $$e_{i,j},\,i=1,2,\ldots,N,\,j=1,2,\ldots,s$$ be $$s$$ independent permutations of the edges $$E(K_n)$$ of the complete graph $$K_n$$. A MultiTree is a set $$I\subseteq [N]$$ such that the edge sets $$E_{I,j}$$ induce spanning trees for $$j=1,2,\ldots,s$$. In this paper we study the following question: what is the smallest $m=m(n)$ such that w.h.p. $[m]$ contains a MultiTree. We prove a hitting time result for $s=2$ and an $$O(n\log n)$$ bound for $$s\geq 3$$. 

Abstract Let be chosen independently and uniformly at random from the unit ‐dimensional cube . Let be given and let . The random geometric graph has vertex set and an edge whenever . We show that if each edge of is colored independently from one of colors and has the smallest value such that has minimum degree at least two, then contains a rainbow Hamilton cycle asymptotically almost surely.