2,699 research outputs found
Amid.cero9 on the Representation of Architecture: Efrén G Grinda and Cristina Díaz Moreno in conversation with Anne Elisabeth Toft and Christina Capetillo
No full textAnother step towards proving a conjecture by Plummer and Toft
No full textAbstractA cyclic colouring of a graph G embedded in a surface is a vertex colouring of G in which any two distinct vertices sharing a face receive distinct colours. The cyclic chromatic number χc(G) of G is the smallest number of colours in a cyclic colouring of G. Plummer and Toft in 1987 [M.D. Plummer, B. Toft, Cyclic coloration of 3-polytopes, J. Graph Theory 11 (1987) 507–515] conjectured that χc(G)≤Δ∗+2 for any 3-connected plane graph G with maximum face degree Δ∗. It is known that the conjecture holds true for Δ∗≤4 and Δ∗≥24. The validity of the conjecture is proved in the paper for Δ∗≥18
Exhibition of oil paintings and water-colour drawings of Newfoundland, and some English pictures
No full textThis is a catalogue from an exhibition of pieces by artist Alfonso Toft, held at Walker's Galleries, London, in 1920. Included in the exhibition are English landscapes and works created from Toft's expedition to Newfoundland. Toft was the guest of Lord Rothmere, who was instrumental in the creation of several paper mills in Newfoundland, and so the dominating subjects of Toft's Newfoundland pieces are the paper mills and surrounding landscapes
Unsolved graph colouring problems
No full textOur book Graph Coloring Problems [85] appeared in 1995. It contains descriptions of unsolved problems, organized into sixteen chapters. A large number of publications on graph colouring have appeared since then, and in particular around thirty of the 211 problems in that book have been solved. In this chapter we review some of our favourite problems that remain unsolved. Introduction A main reason for the continued interest in the area of graph colouring is its wealth of interesting unsolved problems. Many of these are easy to state, but seemingly difficult to solve. However they are not impossible, as the literature in the field will testify. The seven most striking results of the past twenty years are: • the 5-list-colourability of planar graphs (dating back to V. G. Vizing in 1975 and to P. Erdős, A. L. Rubin and H. Taylor in 1979) by Thomassen [159] • the confirmation by Robertson, Sanders, Seymour and Thomas [137] of the truth of the four-colour theorem (F. Guthrie and A. De Morgan (1852)) • the asymptotic solution by Reed [134] of the problem as to whether for k ≥ 9 there are k-chromatic graphs without complete k-graphs and of maximum degree k (V. G. Vizing (1968) and O. V. Borodin and A. V. Kostochka (1977)) • the proof by Chudnovsky, Robertson, Seymour and Thomas [39] of the strong perfect graph conjecture of C. Berge around 1960 • the proof by Thomassen [161] of the weak 3-flow conjecture of W. T. Tutte (1954) and F. Jaeger (1988) • the solution by Kostochka and Yancey [111] to the problem of critical graphs with few edges (due to T. Gallai (1963) and O. Ore (1967)) • the description found by Borodin, Dvořák, Kostochka, Lidický and Yancey [24] of all 4-critical planar graphs with exactly four triangles (B. Grünbaum (1963), V. A. Aksenov (1974) and P. Erdős (1990)). In addition to these major achievements there are many other important results – in fact, thirty-one of the original 211 problems from the lists in Jensen and Toft [85] were solved by 2013.</p
Even cycles in graphs
No full textLet G be a 3-connected simple graph of minimum degree 4 on at least six vertices. The author proves the existence of an even cycle C in G such that G-V ( C ) is connected and G-E ( C ) is 2-connected. The result is related to previous results of Jackson, and Thomassen and Toft. Thomassen and Toft proved that G contains an induced cycle C such that both G-V ( C ) and G-E ( C ) is 2-connected. G does not in general contain an even cycle such that G-V ( C ) is 2-connected. © 2004 Wiley Periodicals, Inc. J Graph Theory 45: 163–223, 2004Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/34893/1/10156_ftp.pd
Non-separating induced cycles in graphs
No full textIn this paper we consider non-separating induced cycles in graphs. A basic result is that any 2-connected graph with at least six vertices and without such a cycle has at least four vertices of degree 2, and this is best possible. For any 3-connected graph G we prove that there exists a non-separating induced cycle C, such that all cycles in G-V(C) are contained in the same block of G-V(C). We apply our results in various directions. In particular, we obtain an extension of a conjecture of Hobbs (first proved by Jackson), and a new proof of Tutte's theorem on 3-connected graphs. Moreover, we show that any graph with minimum degree at least 3 contains a subdivision of K4 in which the three edges of a Hamiltonian path of the K4 are left undivided. This is an extension of a conjecture by Toft and implies an extension of a conjecture of Bollobás and Erdös (first proved by Larson) on the existence of an odd cycle with at least one diagonal. Finally, we obtain a result on the existence of a vertex joined by edges to three vertices of a cycle in a graph. This implies an extremal result conjectured by Bollobás and Erdös (first proved by Thomassen), as well as the conjecture of Toft that every 4-chromatic graph contains such a configuration
Pseudo-differential operators with isotropic symbols, Wick and anti-Wick operators, and hypoellipticity
No full textWe study the link between ilidos and Wick operators via the Bargmann transform. We deduce a formula for the symbol of the Wick operator in terms of the short-time Fourier transform of the Weyl symbol. This gives characterizations of Wick symbols of ilidos of Shubin type and of infinite order, and results on composition. We prove a series expansion of Wick operators in terms of anti-Wick operators which leads to a sharp Garding inequality and transition of hypoellipticity between Wick and Shubin symbols. Finally we show continuity results for anti-Wick operators, and estimates for the Wick symbols of anti-Wick operators.(c) 2022 The Author(s). Published by Elsevier Masson SAS. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/)
On the Hajós number of graphs
No full textAbstractA graph G is said to have property Pm if it contains no subdivision of Km+1 and no subdivision of K⌈m/2⌉+1,⌊m/2⌋+1. Chartrand et al. (J. Combin Theory 10 (1971) 12–41) (see also Problem 6.3 in Jensen and Toft (Graph Coloring Problems, Wiley, New York, 1995) conjectured that the set of vertices (respectively, edges) of any graph with property Pm can be partitioned into m−n+1 subsets such that each of these subsets induces a graph with property Pn, provided m⩾n⩾1 (respectively, m⩾n⩾2). We prove that both conjectures fail when m>cn2 for some positive constant c. In fact, we prove that under the condition m>cn2, there exists a graph G with property Pm such that in every colouring of its vertices or edges with m colours there is a monochromatic subgraph H with Hajós number h(H)>n, that is, with a subdivision of Kn+1. In addition, we prove bounds of Nordhaus–Gaddum type for the Hajós number
Coloring hypergraphs of low connectivity
No full textFor a hypergraph G, let χ(G), Δ(G), and λ(G) denote the chromatic number, the maximum degree, and the maximum local edge connectivity of G, respectively. A result of Rhys Price Jones from 1975 says that every connected hypergraph G satisfies χ(G)≤Δ(G)+1 and equality holds if and only if G is a complete graph, an odd cycle, or G has just one (hyper-)edge. By a result of Bjarne Toft from 1970 it follows that every hypergraph G satisfies χ(G)≤λ(G)+1. In this paper, we show that a hypergraph G with λ(G)≥3 satisfies χ(G)=λ(G)+1 if and only if G contains a block which belongs to a family Hλ(G). The class H3 is the smallest family which contains all odd wheels and is closed under taking Hajós joins. For k≥4, the family Hk is the smallest that contains all complete graphs Kk+1 and is closed under Hajós joins. For the proofs of the above results we use critical hypergraphs. A hypergraph G is called (k+1)-critical if χ(G)=k+1, but χ(H)≤k whenever H is a proper subhypergraph of G. We give a characterization of (k+1)-critical hypergraphs having a separating edge set of size k as well as a a characterization of (k+1)-critical hypergraphs having a separating vertex set of size 2
Présence d'un Asellide épigé originaire d'Extrême-Orient en Californie
No full textPresence of a Far-East epigean Asellid in California
The Asellid Asellus (Asellus) hilgendorfii Bovallius, 1886 was discovered in the calm waters of the Sacramento/San Joaquim River Delta, Californie, USA. The normal range of this species is eastern Siberia, China, and the entire Japanese archipelago, which inclines to the belief that it is a question of a recent introduction, without doubt of human origin.L'Asellidae Asellus (Asellus ) hilgendorfii Bovallius, 1886 a été découvert dans les eaux douces de la région du delta des fleuves Sacramento et San Joaquin, Californie, USA. L'aire de vie normale de cette espèce étant la Sibérie orientale, la Chine et l'ensemble de l'archipel nippon, tout porte à croire qu'il s'agit d'une introduction récente, sans doute d'origine humaine.Magniez Guy, Toft Jason. Présence d'un Asellide épigé originaire d'Extrême-Orient en Californie. In: Bulletin mensuel de la Société linnéenne de Lyon, 69ᵉ année, n°6, juin 2000. pp. 127-132
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