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A graph where each vertex $v$ has a list $L(v)$ of available colors is $L$-colorable if there is a proper coloring such that the color of $v$ is in $L(v)$ for each $v$. A graph is $k$-choosable if every assignment $L$ of at least $k$ colors to each vertex guarantees an $L$-coloring. Given a list assignment $L$, an $L$-request for a vertex $v$ is a color $c\in L(v)$. In this paper, we look at a variant of the widely studied class of precoloring extension problems from [Z. Dvo\v{r}\'ak, S. Norin, and L. Postle: List coloring with requests. J. Graph Theory 2019], wherein one must satisfy "enough", as opposed to all, of the requested set of precolors. A graph $G$ is $\varepsilon$-flexible for list size $k$ if for any $k$-list assignment $L$, and any set $S$ of $L$-requests, there is an $L$-coloring of $G$ satisfying an $\varepsilon$-fraction of the requests in $S$. It is conjectured that planar graphs are $\varepsilon$-flexible for list size $5$, yet it is proved only for list size $6$ and for certain subclasses of planar graphs. We give a stronger version of the main tool used in the proofs of the aforementioned results. By doing so, we improve upon a result by Masa\v{r}\'ik and show that planar graphs without $K_4^-$ are $\varepsilon$-flexible for list size $5$. We also prove that planar graphs without $4$-cycles and $3$-cycle distance at least 2 are $\varepsilon$-flexible for list size $4$. Finally, we introduce a new (slightly weaker) form of $\varepsilon$-flexibility where each vertex has exactly one request. In that setting, we provide a stronger tool and we demonstrate its usefulness to further extend the class of graphs that are $\varepsilon$-flexible for list size $5$., Comment: 18 pages, 4 figures

Introduction Plasma thrombin generation kinetics as measured by the calibrated automated thrombogram (CAT) assay is a predictor of symptomatic venous thromboembolism after trauma. We hypothesized that data from a new prototype assay for measurement of thrombin generation kinetics in fresh whole blood (near patient testing of thrombin generation), will correlate with the standard CAT assay in the same patients, making it a potential tool in the future care of trauma patients. Methods Patients were enrolled from June 2018 to February 2020. Within 12 hours of injury, blood samples were collected simultaneously for both assays. Variables compared and correlated between assays were lag time, peak height, time to peak, and endogenous thrombin potential. Data are presented as median with interquartile range (IQR). Spearman and Pearson correlations were estimated and tested between both assays; a P value of

Patient-centered care requires that treatments respond to the problematic situation of each patient in a manner that makes intellectual, emotional, and practical sense, an achievement that requires shared decision making (SDM). To implement SDM in practice, tools – sometimes called conversation aids or decision aids – are prepared by collating, curating, and presenting high quality, comprehensive, and up-to-date evidence. Yet, the literature offers limited guidance for how to make evidence support SDM. Here, we describe our approach and the challenges encountered during the development of Anticoagulation Choice, a conversation aid to help patients with atrial fibrillation and their clinicians jointly consider the risk of thromboembolic stroke and decide whether and how to respond to this risk with anticoagulation.

In 1984, Matthews and Sumner conjectured that every 4-connected, claw-free graph contains a Hamiltonian cycle. This still unresolved conjecture has been the motivation for research into the existence of other cycle structures. In this paper, we consider the stronger property of pancyclicity for 4-connected graphs. In particular, we show that every 4-connected, { K 1 , 3 , N ( i , j , k ) } -free graph, where i , j , k ≥ 1 and i + j + k = 6 , is pancyclic. This, together with results by Ferrara, Morris, Wenger, and Ferrara et al. completes a characterization of the graphs Y such that every { K 1 , 3 , Y } -free graph is pancyclic. In addition, this represents the best known progress towards answering a question of Gould concerning a characterization of the pairs of forbidden subgraphs that imply pancyclicity in 4-connected graphs.

A 2-factor in a graph is a spanning 2-regular subgraph, or equivalently a spanning collection of disjoint cycles. In this paper we investigate the existence of 2-factors with a bounded number of odd cycles in a graph. We extend results of Ryjacek, Saito, and Schelp (1999) and show that the number of odd components of a 2-factor in a claw-free graph is stable under Ryjacek's closure operation. We also consider conditions that ensure the existence of a pair of disjoint 1-factors in a claw-free graph, as the union of such a pair is a 2-factor with no odd cycles.

Given a family of graphs F, a graph G is F-saturated if no element of F is a subgraph of G, but for any edge e in G@?, some element of F is a subgraph of G+e. Let sat(n,F) denote the minimum number of edges in an F-saturated graph of order n, which we refer to as the saturation number or saturation function of F. If F={F}, then we instead say that G is F-saturated and write sat(n,F). For graphs G,H"1,...,H"k, we write that G->(H"1,...,H"k) if every k-coloring of E(G) contains a monochromatic copy of H"i in color i for some i. A graph G is (H"1,...,H"k)-Ramsey-minimal if G->(H"1,...,H"k) but for any e@?G, (G-e)@?->(H"1,...,H"k). Let R"m"i"n(H"1,...,H"k) denote the family of (H"1,...,H"k)-Ramsey-minimal graphs. In this paper, motivated in part by a conjecture of Hanson and Toft (1987), we prove that sat(n,R"m"i"n(m"1K"2,...,m"kK"2))=3(m"1+...+m"k-k) for m"1,...,m"k>=1 and n>3(m"1+...+m"k-k), and we also characterize the saturated graphs of minimum size. The proof of this result uses a new technique, iterated recoloring, which takes advantage of the structure of H"i-saturated graphs to determine the saturation number of R"m"i"n(H"1,...,H"k).

BackgroundNew‐onset diabetes mellitus after a pancreaticoduodenectomy (PD) remains poorly defined. The aim of this study was to define the incidence and predictive factors of immediate post‐resection diabetes mellitus (iPRDM).MethodsRetrospective review of patients undergoing PD from January 2004 through to July 2010. Immediate post‐resection diabetes mellitus was defined as diabetes requiring pharmacological treatment within 30 days post‐operatively. Logistic regression was conducted to identify factors predictive of iPRDM.ResultsOf 778 patients undergoing PD, 214 were excluded owing to pre‐operative diabetes (n= 192), declined research authorization (n= 14) or death prior to hospital discharge (n= 8); the remaining 564 patients comprised the study population. iPRDM occurred in 22 patients (4%) who were more likely to be male, have pre‐operative glucose intolerance, or an increased creatinine, body mass index (BMI), pre‐operative glucose, operative time, tumour size or specimen length compared with patients without iPRDM (P < 0.05). On multivariate analysis, pre‐operative impaired glucose intolerance (P < 0.001), pre‐operative glucose ≥ 126 (P < 0.001) and specimen length (P= 0.002) were independent predictors of iPRDM. A predictive model using these three factors demonstrated a c‐index of 0.842.DiscussionNew‐onset, post‐resection diabetes occurs in 4% of patients undergoing PD. Factors predictive of iPRDM include pre‐operative glucose intolerance, elevated pre‐operative glucose and increased specimen length. These data are important for patient education and predicting outcomes after PD.

A graph G is pancyclic if it contains cycles of each length l , 3 ≤ l ≤ | V ( G ) | . The generalized bull B ( i , j ) is obtained by associating one endpoint of each of the paths P i + 1 and P j + 1 with distinct vertices of a triangle. Gould, Łuczak and Pfender (2004) [4] showed that if G is a 3-connected { K 1 , 3 , B ( i , j ) } -free graph with i + j = 4 then G is pancyclic. In this paper, we prove that every 4-connected, claw-free, B ( i , j ) -free graph with i + j = 6 is pancyclic. As the line graph of the Petersen graph is B ( i , j ) -free for any i + j = 7 and is not pancyclic, this result is best possible.

We present a systematic approach to selecting cluster heads (CHs) in networked sensing systems for data aggregation based on the idea of dominating sets of a graph. A set of vertices, D of a graph G=(V,E) is a k-dominating set if every vertex v@?V, is connected to some vertex in D via path of length less than or equal to k. By focusing on regular mesh topologies, in which each node has q neighbors, q=3,4,6,8, we demonstrate an intuitive approach for identifying the CHs. If E is the energy required for a node to send a message to a collection point, e is the energy required to send a similar message to its adjacent node in the system, and C is the compression ratio achieved by aggregation techniques at a CH, we show that CHs selected in our approach are viable whenever k

With the increasing complexity of the power amplifier (PA) module architecture, the probability of a thermally induced stress related failure mechanism increases. To help evaluate the increase in module complexity, a more sophisticated in situ monitored thermal cycle reliability test is available. The module is monitored in real time using a resistance daisy chain methodology designed to provide coverage using resistance feedback throughout the entire hierarchy of the module and carrier board interface. Monitored temperature cycling allows for real time failure feedback and enhanced failure signature information. Further, the testing technique has proven to be a valuable method for capturing the early stages of a module mechanical failure at the temperature extremes. Moreover, statistical evaluation of the failure data (Weibull analysis) is improved and better accuracy of the failures in time (FIT) rate can be determined.

It is an elementary exercise to show that any non-trivial simple graph has two vertices with the same degree. This is not the case for digraphs and multigraphs. We consider generating irregular digraphs from arbitrary digraphs by adding multiple arcs. To this end, we define an irregular labeling of a digraph D to be an arc-labeling of the digraph such that the ordered pairs of the sums of the in-labels and out-labels at each vertex are all distinct. We define the strength s→(D) of D to be the smallest of the maximum labels used across all irregular labelings. Similar definitions for graphs have been studied extensively and a different formulation of digraph irregularity was given in [H. Hackett, Irregularity strength of graphs and digraphs, Masters Thesis, University of Louisville, 1995]. Here we continue the study of irregular labelings of digraphs. We give a general lower bound on s→(D) and determine s→(D) exactly for tournaments, directed paths and cycles and the orientation of the path where all vertices have either in-degree 0 or out-degree 0. We also determine the irregularity strength of a union of directed cycles and a union of directed paths, the latter which requires a new result pertaining to finding circuits of given lengths containing prescribed vertices in the complete symmetric digraph with loops.

A deBruijn sequence of orderk, or a k-deBruijn sequence, over an alphabet A is a sequence of length |A|^k in which the last element is considered adjacent to the first and every possible k-tuple from A appears exactly once as a string of k-consecutive elements in the sequence. We will say that a cyclic sequence is deBruijn-like if for some k, each of the consecutive k-element substrings is distinct. A vertex coloring @g:V(G)->[k] of a graph G is said to be proper if no pair of adjacent vertices in G receive the same color. Let C(v;@g) denote the multiset of colors assigned by a coloring @g to the neighbors of vertex v. A proper coloring @g of G is irregular if @g(u)=@g(v) implies that C(u;@g) C(v;@g). The minimum number of colors needed to irregularly color G is called the irregular chromatic number of G. The notion of the irregular chromatic number pairs nicely with other parameters aimed at distinguishing the vertices of a graph. In this paper, we demonstrate a connection between the irregular chromatic number of cycles and the existence of certain deBruijn-like sequences. We then determine the exact irregular chromatic number of C"n and P"n for n>=3, thus verifying two conjectures given by Okamoto, Radcliffe and Zhang.

A graph G is H-saturated if G does not contain H as a subgraph but for any nonadjacent vertices u and v, G+uv contains H as a subgraph. The parameter sat(H,n) is the minimum number of edges in an H-saturated graph of order n. In this paper, we determine sat(H,n) for sufficiently large n when H is a union of cliques of the same order, an arbitrary union of two cliques and a generalized friendship graph.

A nonincreasing sequence of nonnegative integers @p=(d"1,d"2,...,d"n) is graphic if there is a (simple) graph G of order n having degree sequence @p. In this case, G is said to realize@p. For a given graph H, a graphic sequence @p is potentiallyH-graphic if there is some realization of @p containing H as a (weak) subgraph. Let @s(@p) denote the sum of the terms of @p. For a graph H and n@?Z^+, @s(H,n) is defined as the smallest even integer m so that every n-term graphic sequence @p with @s(@p)>=m is potentially H-graphic. Let K"s^t denote the complete t partite graph such that each partite set has exactly s vertices. We show that @s(K"s^t,n)=@s(K"("t"-"2")"s+K"s","s,n) and obtain the exact value of @s(K"j+K"s","s,n) for n sufficiently large. Consequently, we obtain the exact value of @s(K"s^t,n) for n sufficiently large.

Let @w"0(G) denote the number of odd components of a graph G. The deficiency of G is defined as def(G)=max"X"@?"V"("G")(@w"0(G-X)-|X|), and this equals the number of vertices unmatched by any maximum matching of G. A subset X@?V(G) is called a Tutte set (or barrier set) of G if def(G)=@w"0(G-X)-|X|, and an extreme set if def(G-X)=def(G)+|X|. Recently a graph operator, called the D-graph D(G), was defined that has proven very useful in examining Tutte sets and extreme sets of graphs which contain a perfect matching. In this paper we give two natural and related generalizations of the D-graph operator to all simple graphs, both of which have analogues for many of the interesting and useful properties of the original.

We consider a variation of the classical Turan-type extremal problem as introduced by Erdős, Jacobson and Lehel in [Erdős, P., M. Jacobson, and J. Lehel, Graphs realizing the same degree sequence and their respective clique numbers, Graph Theory, Combinatorics and Applications, 1, 1991, ed. Alavi, Chartrand, Oellerman and Schwenk, 439–449]. Let π be an n-element graphic sequence and let H be a graph. We wish to determine the smallest even integer m such that any n-term graphic sequence π having degree sum at least m has some realization containing H as a subgraph. Denote this value m by σ ( H , n ) . For an arbitrarily chosen H, we construct a graphic sequence π ( H , n ) whose degree sum plus two is at least σ ( H , n ) . Furthermore, we conjecture that equality holds in general, as this is the case for all choices of H where σ ( H , n ) is currently known.

for relevant studies (Search date Sept 2013). Case control studies with gastrectomy for benign diseases were included. Gastrectomy for malignant conditions was excluded. Standardized forms were used to extract data. Data was extracted by two reviewers independently. Odds ratio was calculated using comprehensive Meta-analysis software. Newcastle-Ottawa Scale (NOS) was used to assess the quality of study. Heterogeneity was also assessed. Random effects model was used for pooling the data. Results: Eleven case-control studies were included in this study (3951 cases/9609 control). Studies were reported from USA (6), Europe (4) and Taiwan (1). Seven studies included hospital based controls and 4 included population based controls. The sample sized ranged from 142 to 720 (cases) and 180 to 2098 (control). The age of the patients ranged from 20 79 yrs. The risk of pancreatic cancer was not increased in patients with prior history of gastrectomy (OR 1.39; 95% CI, 0.95 -2.03, p= 0.09). Subgroup analysis based on the geographical location revealed the risk was not increased in the USA (OR 1.65; 95% CI, 0.96-2.82, p=0.068) or Europe (OR 1.13; 95% CI 0.61-2.08, p=0.07). Subgroup analysis for Asia was not performed as there was only one study. Subgroup analysis based on the design of the study (population based control or hospital based controls) showed no significant difference. As per NewcastleOttawa Scale, nine studies were of high quality (7 or more NOS points) and two were of low quality (6 NOS points). Subgroup analysis based on quality did not show any significant difference. Moderate heterogeneity was present (I square =56%) and thus random effects model was used for analysis. Conclusion: A prior history of gastrectomy for benign gastric disorders is not associated with an increased risk for pancreatic cancer.

Background: Diabetes is a known, but poorly understood sequela of pancreatic resection in a subset of patients. Our aim was to define the incidence and risk factors of postresection diabetes (PRD) after distal pancreatectomy (DP). Methods: A retrospective review of all consecutive patients undergoing DP at our institution from January 2004 through July 2010. Data was obtained from medical records, pathology reports, and postoperative imaging. Postoperative CT scans were evaluated to determine extent of resection, defined by site of transection as limited (adrenal or left), standard (superior mesenteric artery or left to adrenal), extended (neck), subtotal (into head). Comparisons between patients with and without PRD were evaluated using chi-square, Fisher's exact, and Wilcoxon rank sum tests. Univariate and multivariable associations with PRD were evaluated using logistic regression models and summarized with odds ratios and 95% confidence intervals. Results: Of 583 patients undergoing DP, 90 patients with pre-existing type I (n=24) or type II (n=66) diabetes were excluded. The remaining 493 patients represent the study population with a mean age of 57 years and BMI (kg/m2) of 27.4. Operative indications included benign lesions in 253 patients (51%) and malignancy in 240 (49%). Extent of pancreatic resection was limited in 141 patients (37%), standard in 87 (23%), extended in 128 (33%), and subtotal in 30 (8%). Postoperatively, 179 patients (36%) required no postoperative treatment, 269 (55%) required perioperative hyperglycemic control, and 45 (9%) developed diabetes including 34 requiring insulin, and 11 oral hypoglycemic medication. Patients developing PRD had higher mean preoperative glucose levels (98.2 vs. 112.1, p