An Algebraic Approach to the Non-chromatic Adherence of the DP Color Function

DP-coloring (or correspondence coloring) is a generalization of list coloring that has been widely studied since its introduction by Dvo\v{r}\'{a}k and Postle in 2015. As the analogue of the chromatic polynomial of a graph $G$, $P(G,m)$, the DP color function of $G$, denoted by $P_{DP}(G,m)$, counts the minimum number of DP-colorings over all possible $m$-fold covers. A function $f$ is chromatic-adherent if for every graph $G$, $f(G,a) = P(G,a)$ for some $a \geq \chi(G)$ implies that $f(G,m) = P(G,m)$ for all $m \geq a$. It is known that the DP color function is not chromatic-adherent, but there are only two known graphs that demonstrate this. Suppose $G$ is an $n$-vertex graph and $\mathcal{H}$ is a 3-fold cover of $G$, in this paper we associate with $\mathcal{H}$ a polynomial $f_{G, \mathcal{H}} \in \mathbb{F}_3[x_1, \ldots, x_n]$ so that the number of non-zeros of $f_{G, \mathcal{H}}$ equals the number of $\mathcal{H}$-colorings of $G$. We then use a well-known result of Alon and F\"{u}redi on the number of non-zeros of a polynomial to establish a non-trivial lower bound on $P_{DP}(G,3)$ when $2n > |E(G)|$. Finally, we use this bound to show that there are infinitely many graphs that demonstrate the non-chromatic-adherence of the DP color function.Comment: 8 pages. arXiv admin note: text overlap with arXiv:2107.08154, arXiv:2110.0405

Resting-State Bra in and the FTO Obesity Risk Allele : Default Mode, Sensorimotor, and Salience Network Connectivity Underlying Different Somatosensory Integration and Reward Processing between Genotypes

Single-nucleotide polymorphisms (SNPs) of the fat mass and obesity associated (FTO) gene are linked to obesity, but how these SNPs influence resting-state neural activation is unknown. Few brain-imaging studies have investigated the influence of obesity-related SNPs on neural activity, and no study has investigated resting-state connectivity patterns. We tested connectivity within three, main resting-state networks: default mode (DMN), sensorimotor (SMN), and salience network (SN) in 30 male participants, grouped based on genotype for the rs9939609 FTO SNP, as well as punishment and reward sensitivity measured by the Behavioral Inhibition (BIS) and Behavioral Activation System (BAS) questionnaires. Because obesity is associated with anomalies in both systems, we calculated a BIS/BAS ratio (BBr) accounting for features of both scores. A prominence of BIS over BAS (higher BBr) resulted in increased connectivity in frontal and paralimbic regions. These alterations were more evident in the obesity-associated AA genotype, where a high BBr was also associated with increased SN connectivity in dopaminergic circuitries, and in a subnetwork involved in somatosensory integration regarding food. Participants with AA genotype and high BBr, compared to corresponding participants in the TT genotype, also showed greater DMN connectivity in regions involved in the processing of food cues, and in the SMN for regions involved in visceral perception and reward-based learning. These findings suggest that neural connectivity patterns influence the sensitivity toward punishment and reward more closely in the AA carriers, predisposing them to developing obesity. Our work explains a complex interaction between genetics, neural patterns, and behavioral measures in determining the risk for obesity and may help develop individually-tailored strategies for obesity prevention.De två första författarna delar förstaförfattarskapet.</p