Positional games on random graphs

We introduce and study Maker/Breaker-type positional games on random graphs. Our main concern is to determine the threshold probability $p_{F}$ for the existence of Maker's strategy to claim a member of $F$ in the unbiased game played on the edges of random graph $G(n,p)$, for various target families $F$ of winning sets. More generally, for each probability above this threshold we study the smallest bias $b$ such that Maker wins the $(1\:b)$ biased game. We investigate these functions for a number of basic games, like the connectivity game, the perfect matching game, the clique game and the Hamiltonian cycle game

On the Role of External Constraints in a Spatially Extended Evolutionary Prisoner's Dilemma Game

We study the emergency of mutual cooperation in evolutionary prisoner's dilemma games when the players are located on a square lattice. The players can choose one of the three strategies: cooperation (C), defection (D) or "tit for tat" (T), and their total payoffs come from games with the nearest neighbors. During the random sequential updates the players adopt one of their neighboring strategies if the chosen neighbor has higher payoff. We compare the effect of two types of external constraints added to the Darwinian evolutionary processes. In both cases the strategy of a randomly chosen player is replaced with probability P by another strategy. In the first case, the strategy is replaced by a randomly chosen one among the two others, while in the second case the new strategy is always C. Using generalized mean-field approximations and Monte Carlo simulations the strategy concentrations are evaluated in the stationary state for different strength of external constraints characterized by the probability P.Comment: 19 pages, 10 figure

Hamilton cycles in highly connected and expanding graphs

In this paper we prove a sufficient condition for the existence of a Hamilton cycle, which is applicable to a wide variety of graphs, including relatively sparse graphs. In contrast to previous criteria, ours is based on only two properties: one requiring expansion of ``small'' sets, the other ensuring the existence of an edge between any two disjoint ``large'' sets. We also discuss applications in positional games, random graphs and extremal graph theory.Comment: 19 page

What is Ramsey-equivalent to a clique?

A graph G is Ramsey for H if every two-colouring of the edges of G contains a monochromatic copy of H. Two graphs H and H' are Ramsey-equivalent if every graph G is Ramsey for H if and only if it is Ramsey for H'. In this paper, we study the problem of determining which graphs are Ramsey-equivalent to the complete graph K_k. A famous theorem of Nesetril and Rodl implies that any graph H which is Ramsey-equivalent to K_k must contain K_k. We prove that the only connected graph which is Ramsey-equivalent to K_k is itself. This gives a negative answer to the question of Szabo, Zumstein, and Zurcher on whether K_k is Ramsey-equivalent to K_k.K_2, the graph on k+1 vertices consisting of K_k with a pendent edge. In fact, we prove a stronger result. A graph G is Ramsey minimal for a graph H if it is Ramsey for H but no proper subgraph of G is Ramsey for H. Let s(H) be the smallest minimum degree over all Ramsey minimal graphs for H. The study of s(H) was introduced by Burr, Erdos, and Lovasz, where they show that s(K_k)=(k-1)^2. We prove that s(K_k.K_2)=k-1, and hence K_k and K_k.K_2 are not Ramsey-equivalent. We also address the question of which non-connected graphs are Ramsey-equivalent to K_k. Let f(k,t) be the maximum f such that the graph H=K_k+fK_t, consisting of K_k and f disjoint copies of K_t, is Ramsey-equivalent to K_k. Szabo, Zumstein, and Zurcher gave a lower bound on f(k,t). We prove an upper bound on f(k,t) which is roughly within a factor 2 of the lower bound

On the minimum degree of minimal Ramsey graphs for multiple colours

A graph G is r-Ramsey for a graph H, denoted by G\rightarrow (H)_r, if every r-colouring of the edges of G contains a monochromatic copy of H. The graph G is called r-Ramsey-minimal for H if it is r-Ramsey for H but no proper subgraph of G possesses this property. Let s_r(H) denote the smallest minimum degree of G over all graphs G that are r-Ramsey-minimal for H. The study of the parameter s_2 was initiated by Burr, Erd\H{o}s, and Lov\'{a}sz in 1976 when they showed that for the clique s_2(K_k)=(k-1)^2. In this paper, we study the dependency of s_r(K_k) on r and show that, under the condition that k is constant, s_r(K_k) = r^2 polylog r. We also give an upper bound on s_r(K_k) which is polynomial in both r and k, and we determine s_r(K_3) up to a factor of log r

Optimized Superconducting Nanowire Single Photon Detectors to Maximize Absorptance

Dispersion characteristics of four types of superconducting nanowire single photon detectors, nano-cavity-array- (NCA-), nano-cavity-deflector-array- (NCDA-), nano-cavity-double-deflector-array- (NCDDA-) and nano-cavity-trench-array- (NCTA-) integrated (I-A-SNSPDs) devices was optimized in three periodicity intervals commensurate with half-, three-quarter- and one SPP wavelength. The optimal configurations capable of maximizing NbN absorptance correspond to periodicity dependent tilting in S-orientation (90{\deg} azimuthal orientation). In NCAI-A-SNSPDs absorptance maxima are reached at the plasmonic Brewster angle (PBA) due to light tunneling. The absorptance maximum is attained in a wide plasmonic-pass-band in NCDAI_1/2*lambda-A, inside a flat-plasmonic-pass-band in NCDAI_3/4*lambda-A and inside a narrow plasmonic-band in NCDAI_lambda-A. In NCDDAI_1/2*lambda-A bands of strongly-coupled cavity and plasmonic modes cross, in NCDDAI_3/4*lambda-A an inverted-plasmonic-band-gap develops, while in NCDDAI_lambda-A a narrow plasmonic-pass-band appears inside an inverted-minigap. The absorptance maximum is achieved in NCTAI_1/2*lambda-A inside a plasmonic-pass-band, in NCTAI_3/4*lambda-A at inverted-plasmonic-band-gap center, while in NCTAI_lambda-A inside an inverted-minigap. The highest 95.05% absorptance is attained at perpendicular incidence onto NCTAI_lambda-A. Quarter-wavelength type cavity modes contribute to the near-field enhancement around NbN segments except in NCDAI_lambda-A and NCDDAI_3/4*lambda-A. The polarization contrast is moderate in NCAI-A-SNSPDs (~10^2), NCDAI- and NCDDAI-A-SNSPDs make possible to attain considerably large polarization contrast (~10^2-10^3 and ~10^3-10^4), while NCTAI-A-SNSPDs exhibit a weak polarization selectivity (~10-10^2).Comment: 26 pages, 8 figure

Experimental energy levels and partition function of the 12^{12}C2_2 molecule

The carbon dimer, the $^{12}$C$_2$ molecule, is ubiquitous in astronomical environments. Experimental-quality rovibronic energy levels are reported for $^{12}$C$_2$, based on rovibronic transitions measured for and among its singlet, triplet, and quintet electronic states, reported in 42 publications. The determination utilizes the Measured Active Rotational-Vibrational Energy Levels (MARVEL) technique. The 23,343 transitions measured experimentally and validated within this study determine 5,699 rovibronic energy levels, 1,325, 4,309, and 65 levels for the singlet, triplet, and quintet states investigated, respectively. The MARVEL analysis provides rovibronic energies for six singlet, six triplet, and two quintet electronic states. For example, the lowest measurable energy level of the \astate\ state, corresponding to the $J=2$ total angular momentum quantum number and the $F_1$ spin-multiplet component, is 603.817(5) \cm. This well-determined energy difference should facilitate observations of singlet--triplet intercombination lines which are thought to occur in the interstellar medium and comets. The large number of highly accurate and clearly labeled transitions that can be derived by combining MARVEL energy levels with computed temperature-dependent intensities should help a number of astrophysical observations as well as corresponding laboratory measurements. The experimental rovibronic energy levels, augmented, where needed, with {\it ab initio} variational ones based on empirically adjusted and spin-orbit coupled potential energy curves obtained using the \Duo\ code, are used to obtain a highly accurate partition function, and related thermodynamic data, for $^{12}$C$_2$ up to 4,000 K.Comment: ApJ Supplements (in press), 48 page