On deficiency problems for graphs

Motivated by analogous questions in the setting of Steiner triple systems and Latin squares, Nenadov, Sudakov and Wagner [Completion and deficiency problems, Journal of Combinatorial Theory Series B, 2020] recently introduced the notion of graph deficiency. Given a global spanning property $\mathcal P$ and a graph $G$, the deficiency $\text{def}(G)$ of the graph $G$ with respect to the property $\mathcal P$ is the smallest non-negative integer $t$ such that the join $G*K_t$ has property $\mathcal P$. In particular, Nenadov, Sudakov and Wagner raised the question of determining how many edges an $n$-vertex graph $G$ needs to ensure $G*K_t$ contains a $K_r$-factor (for any fixed $r\geq 3$). In this paper we resolve their problem fully. We also give an analogous result which forces $G*K_t$ to contain any fixed bipartite $(n+t)$-vertex graph of bounded degree and small bandwidth.Comment: 11 page

The induced saturation problem for posets

For a fixed poset $P$, a family $\mathcal F$ of subsets of $[n]$ is induced $P$-saturated if $\mathcal F$ does not contain an induced copy of $P$, but for every subset $S$ of $[n]$ such that $S\not \in \mathcal F$, then $P$ is an induced subposet of $\mathcal F \cup \{S\}$. The size of the smallest such family $\mathcal F$ is denoted by $\text{sat}^* (n,P)$. Keszegh, Lemons, Martin, P\'alv\"olgyi and Patk\'os [Journal of Combinatorial Theory Series A, 2021] proved that there is a dichotomy of behaviour for this parameter: given any poset $P$, either $\text{sat}^* (n,P)=O(1)$ or $\text{sat}^* (n,P)\geq \log _2 n$. In this paper we improve this general result showing that either $\text{sat}^* (n,P)=O(1)$ or $\text{sat}^* (n,P) \geq 2 \sqrt{n-2}$. Our proof makes use of a Tur\'an-type result for digraphs. Curiously, it remains open as to whether our result is essentially best possible or not. On the one hand, a conjecture of Ivan states that for the so-called diamond poset $\Diamond$ we have $\text{sat}^* (n,\Diamond)=\Theta (\sqrt{n})$; so if true this conjecture implies our result is tight up to a multiplicative constant. On the other hand, a conjecture of Keszegh, Lemons, Martin, P\'alv\"olgyi and Patk\'os states that given any poset $P$, either $\text{sat}^* (n,P)=O(1)$ or $\text{sat}^*(n,P)\geq n+1$. We prove that this latter conjecture is true for a certain class of posets $P$.Comment: 12 page

Humanized H19/Igf2 locus reveals diverged imprinting mechanism between mouse and human and reflects Silver–Russell syndrome phenotypes

Genomic imprinting is essential for mammalian development. Curiously, elements that regulate genomic imprinting, the imprinting control regions (ICRs), often diverge across species. To understand whether the diverged ICR sequence plays a species-specific role at the H19/insulin-like growth factor 2 (Igf2) imprinted locus, we generated a mouse in which the human ICR (hIC1) sequence replaced the endogenous mouse ICR. We show that the imprinting mechanism has partially diverged between mouse and human, depending on the parental origin of the hIC1 in mouse. We also suggest that our mouse model is optimal for studying the imprinting disorders Beckwith–Wiedemann syndrome when hIC1 is maternally transmitted, and Silver–Russell syndrome when hIC1 is paternally transmitted

11.Hypoxic ventilatory depressionと思われる一症例について(第551回千葉医学会例会・第9回麻酔科例会・第18回千葉麻酔懇話会)

FISH analysis on metaphase nuclei (top panel) of cultured cells derived from peripheral blood leukocytes of the proband of family 2 by using BAC probes for 11p15.5-15.4 (RP11-11A9, 3,236,552-3,356,012, green) and 11q22.3 (RP11-179B7, 104,298,339-104,459,797, red). The green signal on both homologues is visible only at chr11p, demonstrating the presence of an in cis duplication and excluding an unbalanced translocation. FISH analysis on interphase nuclei (bottom panel) using the BACs RP11-699D10 (2.9–3.0 Mb, red) and RP11-11A9 (green), hybridizing within the duplication. Note that single and duplicated signals can be seen on the two homologues, respectively. The red-green-green-red order of the duplicated signals indicates that the duplication is inverted. (PDF 52 kb