Erd\H{o}s-Gy\'{a}rf\'{a}s Conjecture for P10P_{10}-free Graphs

Let $P_{10}$ be a path on $10$ vertices. A graph is said to be $P_{10}$-free if it does not contain $P_{10}$ as an induced subgraph. The well-known Erd\H{o}s-Gy\'{a}rf\'{a}s Conjecture states that every graph with minimum degree at least three has a cycle whose length is a power of $2$. In this paper, we show that every $P_{10}$-free graph with minimum degree at least three contains a cycle of length $4$ or $8$. This implies that the conjecture is true for $P_{10}$-free graphs

PAV markers in <i>Sorghum bicolour</i>:genome pattern, affected genes and pathways, and genetic linkage map construction

KEY MESSAGE: 5,511 genic small-size PAVs in sorghum were identified and examined, including the pattern and the function enrichment of PAV genes. 325 PAV markers were developed to construct a genetic map. ABSTRACT: Presence/absence variants (PAVs) correlate closely to the phenotypic variation, by impacting plant genome sizes and the adaption to the environment. To shed more light on their genome-wide patterns, functions and the possibility of using them as molecular markers, we generated next generation genome sequencing data for four sorghum inbred lines and used associated bioinformatic pipelines to identify small-size PAVs (40–10 kb). Five thousand five hundreds and eleven genic PAVs (40–10 kb) were identified and found to affect 3,238 genes. These PAVs were mainly distributed on the sub-telomeric regions, but the highest proportions occurred in the vicinity of the centromeric regions. One of the prominent features of the PAVs is the high occurrence of long terminal repeats retrotransposons and DNA transposons. PAVs caused various alterations to gene structure, primarily including the coding sequence variants, intron variants, transcript ablation, and initiator codon changes. The genes affected by PAVs were significantly enriched in those involved in stress responses and protein modification. We used 325 PAVs polymorphic between two sorghum inbred lines Ji2731 and E-Tian, together with 49 SSR markers, and constructed a genetic map, which consisted of 10 linkage groups corresponding to the 10 chromosomes of sorghum and spanned 1,430.3 cM in length covering 97 % of the physical genome. The resources reported here should be useful for genetic study and breeding of sorghum and related species. ELECTRONIC SUPPLEMENTARY MATERIAL: The online version of this article (doi:10.1007/s00122-015-2458-4) contains supplementary material, which is available to authorized users

Quantum diffusion of microcavity solitons

Coherently pumped (Kerr) solitons in an ideal optical microcavity are expected to undergo random quantum motion that determines fundamental performance limits in applications of the soliton microcombs. Here this random walk and its impact on Kerr soliton timing jitter are studied experimentally. The quantum limit is discerned by measuring the relative position of counter-propagating solitons. Their relative motion features weak interactions and also presents common-mode suppression of technical noise, which typically hides the quantum fluctuations. This is in contrast to co-propagating solitons, which are found to have relative timing jitter well below the quantum limit of a single soliton on account of strong correlation of their mutual motion. Good agreement is found between theory and experiment. The results establish the fundamental limits to timing jitter in soliton microcombs and provide new insights on multisoliton physics

Interleaved difference-frequency-generation for mid-infrared microcomb spectral densification

Generation of mid-infrared combs (3.3 micron band) with GigaHertz line spacing is demonstrated by interleaved difference-frequency-generation. The method, applied to a 22 GHz repetition-rate microcomb, is useful for spectral densification of sparse microcomb spectra

Interleaved difference-frequency generation for microcomb spectral densification in the mid-infrared

With their compact size and semiconductor-chip-based operation, frequency microcombs can be an invaluable light source for gas spectrcoscopy. However, the generation of mid-infrared (mid-IR) frequency combs with gigahertz line spacing as required to resolve many gas spectra represents a significant challenge for these devices. Here, a technique referred to as interleaved difference-frequency generation (iDFG) is introduced that densifies the spectral line spacing upon conversion of near-IR comb light into the mid-IR light. A soliton microcomb is used as both a comb light source and microwave oscillator in a demonstration, and the spectrum of methane is measured to illustrate how the resulting mid-IR comb avoids spectral undersampling. Beyond demonstration of the iDFG technique, this work represents an important feasibility step towards more compact and potentially chip-based mid-IR gas spectroscopy modules

Interleaved difference-frequency-generation for mid-infrared microcomb spectral densification

Generation of mid-infrared combs (3.3 micron band) with GigaHertz line spacing is demonstrated by interleaved difference-frequency-generation. The method, applied to a 22 GHz repetition-rate microcomb, is useful for spectral densification of sparse microcomb spectra

Impact of spatio-temporal thermal decoherence on soliton microcombs in multimode microresonators

The phase noise of the soliton repetition rate is experimentally characterized in silica microresonators. In conjunction with dispersive wave quieting of pump technical noise, spatio-temporal fluctuations of distinct transverse modes set a limit to performance

Impact of spatio-temporal thermal decoherence on soliton microcombs in multimode microresonators

The phase noise of the soliton repetition rate is experimentally characterized in silica microresonators. In conjunction with dispersive wave quieting of pump technical noise, spatio-temporal fluctuations of distinct transverse modes set a limit to performance