Critical Photoperiod and Optimal Quality of Night Interruption Light for Runner Induction in June-Bearing Strawberries

The optimal photoperiod and light quality for runner induction in strawberries ‘Sulhyang’ and ‘Maehyang’ were investigated. Two experiments were carried out in a semi-closed walk-in growth chamber with 25/15 °C day/night temperatures and a light intensity of 250 μmol·m–2·s–1photosynthetic photon flux density (PPFD) provided from white light-emitting diodes (LEDs). In the first experiment, plants were treated with a photoperiod of either 12, 14, 16, 18, 20, or 22 h In the second experiment, a total of 4 h of night interruption (NI) light at an intensity of 70 μmol·m–2·s–1PPFD provided from either red, blue, green, white, or far-red LED in addition to 11 h short day (SD). The results showed that both ‘Sulhyang’ and ‘Maehyang’ produced runners when a photoperiod was longer than 16 h, and the number of runners induced positively correlated with the length of photoperiod. However, the plant growth, contents of chlorophyll, sugar and starch, and Fv/Fo decreased in a 22 h photoperiod. All qualities of the NI light, especially red light, significantly increased the number of runners and daughter plants induced per plant as compared with those in the SD treatment in both cultivars. In a conclusion, a photoperiod between 16 and 20 h and NI light, especially red NI light, can be used for quality runner induction in both ‘Sulhyang’ and ‘Maehyang’.

Homomorphisms of Sparse Signed Graphs

The notion of homomorphism of signed graphs, introduced quite recently, provides better interplay with the notion of minor and is thus of high importance in graph coloring. A newer, but equivalent, definition of homomorphisms of signed graphs, proposed jointly by the second and third authors of this paper and Thomas Zaslavsky, leads to a basic no-homomorphism lemma. According to this definition, a signed graph $(G, \sigma)$ admits a homomorphism to a signed graph $(H, \pi)$ if there is a mapping $\phi$ from the vertices and edges of $G$ to the vertices and edges of $H$ (respectively) which preserves adjacencies, incidences, and signs of closed walks (i.e., the product of the sign of their edges). For $ij=00, 01, 10, 11$, let $g_{ij}(G,\sigma)$ be the length of a shortest nontrivial closed walk of $(G, \sigma)$ which is, positive and of even length for $ij=00$, positive and of odd length for $ij=01$, negative and of even length for $ij=10$, negative and of odd length for $ij=11$. For each $ij$, if there is no nontrivial closed walk of the corresponding type, we let $g_{ij}(G, \sigma)=\infty$. If $G$ is bipartite, then $g_{01}(G,\sigma)=g_{11}(G,\sigma)=\infty$. In this case, $g_{10}(G,\sigma)$ is certainly realized by a cycle of $G$, and it will be referred to as the \emph{unbalanced-girth} of $(G,\sigma)$. It then follows that if $(G,\sigma)$ admits a homomorphism to $(H, \pi)$, then $g_{ij}(G, \sigma)\geq g_{ij}(H, \pi)$ for $ij \in \{00, 01,10,11\}$. Studying the restriction of homomorphisms of signed graphs on sparse families, in this paper we first prove that for any given signed graph $(H, \pi)$, there exists a positive value of $\epsilon$ such that, if $G$ is a connected graph of maximum average degree less than $2+\epsilon$, and if $\sigma$ is a signature of $G$ such that $g_{ij}(G, \sigma)\geq g_{ij}(H, \pi)$ for all $ij \in \{00, 01,10,11\}$, then $(G, \sigma)$ admits a homomorphism to $(H, \pi)$. For $(H, \pi)$ being the signed graph on $K_4$ with exactly one negative edge, we show that $\epsilon=\frac{4}{7}$ works and that this is the best possible value of $\epsilon$. For $(H, \pi)$ being the negative cycle of length $2g$, denoted $UC_{2g}$, we show that $\epsilon=\frac{1}{2g-1}$ works. As a bipartite analogue of the Jaeger-Zhang conjecture, Naserasr, Sopena and Rollovà conjectured in [Homomorphisms of signed graphs, {\em J. Graph Theory} 79 (2015)] that every signed bipartite planar graph $(G,\sigma)$ satisfying $g_{ij}(G,\sigma)\geq 4g-2$ admits a homomorphism to $UC_{2g}$. We show that $4g-2$ cannot be strengthened, and, supporting the conjecture, we prove it for planar signed bipartite graphs $(G,\sigma)$ satisfying the weaker condition $g_{ij}(G,\sigma)\geq 8g-2$. In the course of our work, we also provide a duality theorem to decide whether a 2-edge-colored graph admits a homomorphism to a certain class of 2-edge-colored signed graphs or not.

Enumerating Unlabelled Embeddings of Digraphs

A 2-cell embedding of an Eulerian digraph D into a closed surface is said to be directed if the boundary of each face is a directed closed walk in D. In this paper, a method is developed with the purpose of enumerating unlabelled embeddings for an Eulerian digraph. As an application, we obtain explicit formulas for the number of unlabelled embeddings of directed bouquets of cycles Bn, directed dipoles OD2n and for a class of regular tournaments T2n+1.

Power Optimization of Multimode Mobile Embedded Systems with Workload-Delay Dependency

This paper proposes to take the relationship between delay and workload into account in the power optimization of microprocessors in mobile embedded systems. Since the components outside a device continuously change their values or properties, the workload to be handled by the systems becomes dynamic and variable. This variable workload is formulated as a staircase function of the delay taken at the previous iteration in this paper and applied to the power optimization of DVFS (dynamic voltage-frequency scaling). In doing so, a graph representation of all possible workload/mode changes during the lifetime of a device, Workload Transition Graph (WTG), is proposed. Then, the power optimization problem is transformed into finding a cycle (closed walk) in WTG which minimizes the average power consumption over it. Out of the obtained optimal cycle of WTG, one can derive the optimal power management policy of the target device. It is shown that the proposed policy is valid for both continuous and discrete DVFS models. The effectiveness of the proposed power optimization policy is demonstrated with the simulation results of synthetic and real-life examples.