Thomassen [Combinatorica 24 (2004), 699–718] proved that a 2–connected, compact, locally connected metric space is homeomorphic to a subset of the sphere if and only if it does not contain K5 or K3,3. The “thumbtack space” consisting of a disc plus an arc attaching just at the centre of the disc shows the assumption of 2–connectedness cannot be dropped. In this work, we introduce “generalized t...

Scientific Reports 5:13545; doi: 10.1038/srep13545; published online 27 August 2015; updated on 09 February 2016 In the Supplementary Information file originally published with this Article, parameter k6 was omitted from Table 2. This error has been corrected in the Supplementary Information that now accompanies the Article. In addition, the model source codes were omitted. There was also an er...

A well known theorem of Kuratowski states that a graph is planar iff it contains no subdivision of K5 or K3,3. Seymour conjectured in 1977 that every 5-connected nonplanar graph contains a subdivision of K5. In this paper, we prove several results about independent paths (no vertex of a path is internal to another), which are then used to prove Seymour’s conjecture for two classes of graphs. Th...

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In the course of extending Grötzsch’s theorem, we prove that every triangle-free graph without a K5-minor is 3-colorable. It has been recently proved that every triangle-free planar graph admits a homomorphism to the Clebsch graph. We also extend this result to the class of triangle-free graphs without a K5-minor. This is related to some conjectures which generalize the Four-Color Theorem. Whil...

The sequent calculus GK5 (GK5D) for the modal propositional logic K5 (K5D) is presented, and it is shown that, every provable sequent Γ → Θ in GK5 (GK5D) has a GK5-proof (GK5D-proof) such that every formula occurring in it is either a subformula of some formulas in Γ, Θ, or the formula 2¬2B or ¬2B, where 2B occurs in the scope of some occurrence of 2 in some formulas of Γ, Θ. Some corollaries i...

The lytic salmonella phage SP6 encodes a tail protein with a high degree of sequence similarity to the tail protein of the biologically unrelated lysogenic salmonella phage P22. The SP6 tail gene is flanked by an upstream region that contains a promoter and a downstream region that contains a putative Rho-independent transcription terminator, giving it a cassette or modular structure almost ide...