Some local--global phenomena in locally finite graphs

In this paper we present some results for a connected infinite graph $G$ with finite degrees where the properties of balls of small radii guarantee the existence of some Hamiltonian and connectivity properties of $G$. (For a vertex $w$ of a graph $G$ the ball of radius $r$ centered at $w$ is the subgraph of $G$ induced by the set $M_r(w)$ of vertices whose distance from $w$ does not exceed $r$). In particular, we prove that if every ball of radius 2 in $G$ is 2-connected and $G$ satisfies the condition $d_G(u)+d_G(v)\geq |M_2(w)|-1$ for each path $uwv$ in $G$, where $u$ and $v$ are non-adjacent vertices, then $G$ has a Hamiltonian curve, introduced by K\"undgen, Li and Thomassen (2017). Furthermore, we prove that if every ball of radius 1 in $G$ satisfies Ore's condition (1960) then all balls of any radius in $G$ are Hamiltonian.Comment: 18 pages, 6 figures; journal accepted versio

Extending partial edge colorings of iterated cartesian products of cycles and paths

We consider the problem of extending partial edge colorings of iterated cartesian products of even cycles and paths, focusing on the case when the precolored edges satisfy either an Evans-type condition or is a matching. In particular, we prove that if $G=C^d_{2k}$ is the $d$th power of the cartesian product of the even cycle $C_{2k}$ with itself, and at most $2d-1$ edges of $G$ are precolored, then there is a proper $2d$-edge coloring of $G$ that agrees with the partial coloring. We show that the same conclusion holds, without restrictions on the number of precolored edges, if any two precolored edges are at distance at least $4$ from each other. For odd cycles of length at least $5$, we prove that if $G=C^d_{2k+1}$ is the $d$th power of the cartesian product of the odd cycle $C_{2k+1}$ with itself ($k\geq2$), and at most $2d$ edges of $G$ are precolored, then there is a proper $(2d+1)$-edge coloring of $G$ that agrees with the partial coloring. Our results generalize previous ones on precoloring extension of hypercubes [Journal of Graph Theory 95 (2020) 410--444]

On star edge colorings of bipartite and subcubic graphs

A star edge coloring of a graph is a proper edge coloring with no $2$-colored path or cycle of length four. The star chromatic index $\chi'_{st}(G)$ of $G$ is the minimum number $t$ for which $G$ has a star edge coloring with $t$ colors. We prove upper bounds for the star chromatic index of complete bipartite graphs; in particular we obtain tight upper bounds for the case when one part has size at most $3$. We also consider bipartite graphs $G$ where all vertices in one part have maximum degree $2$ and all vertices in the other part has maximum degree $b$. Let $k$ be an integer ($k\geq 1$), we prove that if $b=2k+1$ then $\chi'_{st}(G) \leq 3k+2$; and if $b=2k$, then $\chi'_{st}(G) \leq 3k$; both upper bounds are sharp. Finally, we consider the well-known conjecture that subcubic graphs have star chromatic index at most $6$; in particular we settle this conjecture for cubic Halin graphs.Comment: 18 page

Local Conditions for Long Cycles in Graphs

A Hamilton cycle in a graph is a cycle that passes through every vertex of the graph. A graph is called Hamiltonian if it contains such a cycle. The problem of determining if a graph is Hamiltonian has been studied extensively, and there are many known sufficient conditions both for Hamiltonicity and for other, related properties. A large portion of these conditions relate the degrees of vertices of the graph to the number of vertices in the entire graph, and thus they can only apply to a limited set of graphs with high edge density. In a series of papers, Asratian and Khachatryan developed local analogues of some of these criteria. These results do not suffer from the same drawbacks as their global counterparts, and apply to larger classes of graphs. In this thesis we study this approach of creating local conditions for Hamiltonicity and related properties, and use it to develop local analogues of some classic results. We will also see how these local conditions can allow us to extend theorems on Hamiltonicity to infinite graphs.En Hamiltoncykel i en graf är en cykel som passerar genom varje hörn i grafen, och en graph är Hamiltonsk om den innehåller en sådan cykel. Problemet att avgöra om en graf är Hamiltonsk har studerats mycket, och det finns många kända villkor som garanterar Hamiltonicitet och andra relaterade egenskaper.  En stor del av dessa villkor sätter gradtalen för hörn i grafen i relation till antalet hörn i hela grafen, och de kan därför endast tillämpas på en begränsad mängd grafer med hög kanttäthet. I ett antal artiklar utvecklade Asratian och Khachatryan lokala motsvarigheter till några av dessa villkor. Dessa resultat har inte samma nackdelar som deras globala motsvarigheter, och kan appliceras på en större mängd grafer.  I denna avhandling undersöker vi detta tillvägagångssätt att skapa lokala villkor för Hamiltonicitet och relaterade egenskaper, och använder det för att ta fram lokala motsvarigheter till några klassiska resultat. Vi kommer också se hur dessa lokala villkor kan tillåta oss att utvidga Hamiltonicitetssatser till oändliga grafer

Pantoprazole in Patients at Risk for Gastrointestinal Bleeding in the ICU

BACKGROUND Prophylaxis for gastrointestinal stress ulceration is frequently given to patients in the intensive care unit (ICU), but its risks and benefits are unclear. METHODS In this European, multicenter, parallel-group, blinded trial, we randomly assigned adults who had been admitted to the ICU for an acute condition (i.e., an unplanned admission) and who were at risk for gastrointestinal bleeding to receive 40 mg of intravenous pantoprazole (a proton-pump inhibitor) or placebo daily during the ICU stay. The primary outcome was death by 90 days after randomization. RESULTS A total of 3298 patients were enrolled; 1645 were randomly assigned to the pantoprazole group and 1653 to the placebo group. Data on the primary outcome were available for 3282 patients (99.5%). At 90 days, 510 patients (31.1%) in the pantoprazole group and 499 (30.4%) in the placebo group had died (relative risk, 1.02; 95% confidence interval [CI], 0.91 to 1.13; P=0.76). During the ICU stay, at least one clinically important event (a composite of clinically important gastrointestinal bleeding, pneumonia, Clostridium difficile infection, or myocardial ischemia) had occurred in 21.9% of patients assigned to pantoprazole and 22.6% of those assigned to placebo (relative risk, 0.96; 95% CI, 0.83 to 1.11). In the pantoprazole group, 2.5% of patients had clinically important gastrointestinal bleeding, as compared with 4.2% in the placebo group. The number of patients with infections or serious adverse reactions and the percentage of days alive without life support within 90 days were similar in the two groups. CONCLUSIONS Among adult patients in the ICU who were at risk for gastrointestinal bleeding, mortality at 90 days and the number of clinically important events were similar in those assigned to pantoprazole and those assigned to placebo. (Funded by Innovation Fund Denmark and others; SUP-ICU ClinicalTrials.gov number, NCT02467621 .)

Pantoprazole in patients at risk for gastrointestinal bleeding in the ICU

BACKGROUND Prophylaxis for gastrointestinal stress ulceration is frequently given to patients in the intensive care unit (ICU), but its risks and benefits are unclear. METHODS In this European, multicenter, parallel-group, blinded trial, we randomly assigned adults who had been admitted to the ICU for an acute condition (i.e., an unplanned admission) and who were at risk for gastrointestinal bleeding to receive 40 mg of intravenous pantoprazole (a proton-pump inhibitor) or placebo daily during the ICU stay. The primary outcome was death by 90 days after randomization. RESULTS A total of 3298 patients were enrolled; 1645 were randomly assigned to the pantoprazole group and 1653 to the placebo group. Data on the primary outcome were available for 3282 patients (99.5%). At 90 days, 510 patients (31.1%) in the pantoprazole group and 499 (30.4%) in the placebo group had died (relative risk, 1.02; 95% confidence interval [CI], 0.91 to 1.13; P=0.76). During the ICU stay, at least one clinically important event (a composite of clinically important gastrointestinal bleeding, pneumonia, Clostridium difficile infection, or myocardial ischemia) had occurred in 21.9% of patients assigned to pantoprazole and 22.6% of those assigned to placebo (relative risk, 0.96; 95% CI, 0.83 to 1.11). In the pantoprazole group, 2.5% of patients had clinically important gastrointestinal bleeding, as compared with 4.2% in the placebo group. The number of patients with infections or serious adverse reactions and the percentage of days alive without life support within 90 days were similar in the two groups. CONCLUSIONS Among adult patients in the ICU who were at risk for gastrointestinal bleeding, mortality at 90 days and the number of clinically important events were similar in those assigned to pantoprazole and those assigned to placebo. (Funded by Innovation Fund Denmark and others; SUP-ICU ClinicalTrials.gov number, NCT02467621.