Complexity of Coloring Graphs without Paths and Cycles

Let $P_t$ and $C_\ell$ denote a path on $t$ vertices and a cycle on $\ell$ vertices, respectively. In this paper we study the $k$-coloring problem for $(P_t,C_\ell)$-free graphs. Maffray and Morel, and Bruce, Hoang and Sawada, have proved that 3-colorability of $P_5$-free graphs has a finite forbidden induced subgraphs characterization, while Hoang, Moore, Recoskie, Sawada, and Vatshelle have shown that $k$-colorability of $P_5$-free graphs for $k \geq 4$ does not. These authors have also shown, aided by a computer search, that 4-colorability of $(P_5,C_5)$-free graphs does have a finite forbidden induced subgraph characterization. We prove that for any $k$, the $k$-colorability of $(P_6,C_4)$-free graphs has a finite forbidden induced subgraph characterization. We provide the full lists of forbidden induced subgraphs for $k=3$ and $k=4$. As an application, we obtain certifying polynomial time algorithms for 3-coloring and 4-coloring $(P_6,C_4)$-free graphs. (Polynomial time algorithms have been previously obtained by Golovach, Paulusma, and Song, but those algorithms are not certifying); To complement these results we show that in most other cases the $k$-coloring problem for $(P_t,C_\ell)$-free graphs is NP-complete. Specifically, for $\ell=5$ we show that $k$-coloring is NP-complete for $(P_t,C_5)$-free graphs when $k \ge 4$ and $t \ge 7$; for $\ell \ge 6$ we show that $k$-coloring is NP-complete for $(P_t,C_\ell)$-free graphs when $k \ge 5$, $t \ge 6$; and additionally, for $\ell=7$, we show that $k$-coloring is also NP-complete for $(P_t,C_7)$-free graphs if $k = 4$ and $t\ge 9$. This is the first systematic study of the complexity of the $k$-coloring problem for $(P_t,C_\ell)$-free graphs. We almost completely classify the complexity for the cases when $k \geq 4, \ell \geq 4$, and identify the last three open cases

Exhaustive generation of kk-critical H\mathcal H-free graphs

We describe an algorithm for generating all $k$-critical $\mathcal H$-free graphs, based on a method of Ho\`{a}ng et al. Using this algorithm, we prove that there are only finitely many $4$-critical $(P_7,C_k)$-free graphs, for both $k=4$ and $k=5$. We also show that there are only finitely many $4$-critical graphs $(P_8,C_4)$-free graphs. For each case of these cases we also give the complete lists of critical graphs and vertex-critical graphs. These results generalize previous work by Hell and Huang, and yield certifying algorithms for the $3$-colorability problem in the respective classes. Moreover, we prove that for every $t$, the class of 4-critical planar $P_t$-free graphs is finite. We also determine all 27 4-critical planar $(P_7,C_6)$-free graphs. We also prove that every $P_{10}$-free graph of girth at least five is 3-colorable, and determine the smallest 4-chromatic $P_{12}$-free graph of girth five. Moreover, we show that every $P_{13}$-free graph of girth at least six and every $P_{16}$-free graph of girth at least seven is 3-colorable. This strengthens results of Golovach et al.Comment: 17 pages, improved girth results. arXiv admin note: text overlap with arXiv:1504.0697

List coloring in the absence of a linear forest.

The k-Coloring problem is to decide whether a graph can be colored with at most k colors such that no two adjacent vertices receive the same color. The Listk-Coloring problem requires in addition that every vertex u must receive a color from some given set L(u)⊆{1,…,k}. Let Pn denote the path on n vertices, and G+H and rH the disjoint union of two graphs G and H and r copies of H, respectively. For any two fixed integers k and r, we show that Listk-Coloring can be solved in polynomial time for graphs with no induced rP1+P5, hereby extending the result of Hoàng, Kamiński, Lozin, Sawada and Shu for graphs with no induced P5. Our result is tight; we prove that for any graph H that is a supergraph of P1+P5 with at least 5 edges, already List 5-Coloring is NP-complete for graphs with no induced H

DNA adducts in fish following an oil spill exposure

On 12 December 1999, one third of the load of the Erika tanker, amounting to about 10,000 t crude oil flowed into sea waters close to the French Atlantic Coast. This oil contained polycyclic aromatic compounds (PAC) that are known to be genotoxic. Genotoxic effects induce DNA adducts formation, which can thus be used as pollution biomarkers. Here, we assessed the genotoxic impact of the “Erika” oil spill by DNA adducts detection in the liver of immature fishes (Solea solea) from four locations of the French Brittany coasts. Two months after the spill, a high amount of DNA adducts was found in samples from all locations, amounting to 92–290 DNA adduct per 109 nucleotides. Then total DNA adduct levels decreased to reach about 50 adducts per 109 nucleotides nine months after the spill. In vitro experiments using human cell cultures and fish liver microsomes evidence the genotoxicity of the Erika fuel. They also prove the formation of reactive species able to create DNA adducts. Furthermore, in vitro and in vivo DNA adducts fingerprints are similar, thus confirming that DNA adducts are a result of the oil spill

Open problems on graph coloring for special graph classes.

For a given graph G and integer k, the Coloring problem is that of testing whether G has a k-coloring, that is, whether there exists a vertex mapping c:V→{1,2,…}c:V→{1,2,…} such that c(u)≠c(v)c(u)≠c(v) for every edge uv∈Euv∈E. We survey known results on the computational complexity of Coloring for graph classes that are hereditary or for which some graph parameter is bounded. We also consider coloring variants, such as precoloring extensions and list colorings and give some open problems in the area of on-line coloring

Risk of betel chewing for oesophageal cancer in Taiwan

Among 104 cases of squamous-cell oesophageal carcinoma patients and 277 controls in Taiwan, after adjusting for cigarette smoking, alcohol consumption, and other confounders, we found that subjects who chewed from 1 to 495 betel-year and more than 495 betel-years (about 20 betel quid per day for 20 years) had 3.6-fold (95% Cl = 1.3–10.1) and 9.2-fold risk (95% Cl = 1.8–46.7), respectively, of developing oesophageal cancer, compared to those who did not chew betel. © 2001 Cancer Research Campaign http://www.bjcancer.co

Arterial bicarbonate is associated with hypoxic burden and uncontrolled hypertension in obstructive sleep apnea - The ESADA cohort

Objective: Blood bicarbonate concentration plays an important role for obstructive sleep apnea (OSA) patients to maintain acid-base balance. We investigated the association between arterial standard bicarbonate ([HCO3-]) and nocturnal hypoxia as well as comorbid hypertension in OSA. Methods: A cross-sectional analysis of 3329 patients in the European Sleep Apnea Database (ESADA) was performed. Arterial blood gas analysis and lung function test were performed in conjunction with polysomnographic sleep studies. The 4% oxygen desaturation index (ODI), mean and minimum oxygen saturation (SpO2), and percentage of time with SpO2 below 90% (T90%) were used to reflect nocturnal hypoxic burden. Arterial hypertension was defined as a physician diagnosis of hypertension with ongoing antihypertensive medication. Hypertensive patients with SBP/DBP below or above 140/90 mmHg were classified as controlled-, uncontrolled hypertension, respectively. Results: The [HCO3-] level was normal in most patients (average 24.0 ± 2.5 mmol/L). ODI, T90% increased whereas mean and minimum SpO2 decreased across [HCO3-] tertiles (ANOVA, p = 0.030, <0.001, <0.001, and <0.001, respectively). [HCO3-] was independently associated with ODI, mean SpO2, minimum SpO2, and T90% after adjusting for confounders (β value [95%CI]: 1.21 [0.88–1.54], −0.16 [-0.20 to −0.11], −0.51 [-0.64 to −0.37], 1.76 [1.48–2.04], respectively, all p < 0.001). 1 mmol/L elevation of [HCO3-] was associated with a 4% increased odds of uncontrolled hypertension (OR: 1.04 [1.01–1.08], p = 0.013). Conclusion: We first demonstrated an independent association between [HCO3-] and nocturnal hypoxic burden as well as uncontrolled hypertension in OSA patients. Bicarbonate levels as an adjunctive measure provide insight into the pathophysiology of hypertension in OSA