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1.
Christian Bosse 《Discrete Mathematics》2019,342(12):111595
The Hadwiger number of a graph , denoted , is the largest integer such that contains as a minor. A famous conjecture due to Hadwiger in 1943 states that for every graph , , where denotes the chromatic number of . Let denote the independence number of . A graph is -free if it does not contain the graph as an induced subgraph. In 2003, Plummer, Stiebitz and Toft proved that for all -free graphs with , where is any graph on four vertices with , , or is a particular graph on seven vertices. In 2010, Kriesell subsequently generalized the statement to include all forbidden subgraphs on five vertices with . In this note, we prove that for all -free graphs with , where denotes the wheel on six vertices. 相似文献
2.
We examine two criteria for balance of a gain graph, one based on binary cycles and one on circles. The graphs for which each criterion is valid depend on the set of allowed gain groups. The binary cycle test is invalid, except for forests, if any possible gain group has an element of odd order. Assuming all groups are allowed, or all abelian groups, or merely the cyclic group of order 3, we characterize, both constructively and by forbidden minors, the graphs for which the circle test is valid. It turns out that these three classes of groups have the same set of forbidden minors. The exact reason for the importance of the ternary cyclic group is not clear. © 2005 Wiley Periodicals, Inc. J Graph Theory 相似文献
3.
Brian Lucena 《Discrete Applied Mathematics》2007,155(8):1055-1065
One consequence of the graph minor theorem is that for every k there exists a finite obstruction set Obs(TW?k). However, relatively little is known about these sets, and very few general obstructions are known. The ones that are known are the cliques, and graphs which are formed by removing a few edges from a clique. This paper gives several general constructions of minimal forbidden minors which are sparse in the sense that the ratio of the treewidth to the number of vertices n does not approach 1 as n approaches infinity. We accomplish this by a novel combination of using brambles to provide lower bounds and achievable sets to demonstrate upper bounds. Additionally, we determine the exact treewidth of other basic graph constructions which are not minimal forbidden minors. 相似文献
4.
In the core of the seminal Graph Minor Theory of Robertson and Seymour lies a powerful theorem capturing the ``rough' structure
of graphs excluding a fixed minor. This result was used to prove Wagner's Conjecture that finite graphs are well-quasi-ordered
under the graph minor relation. Recently, a number of beautiful results that use this structural result have appeared. Some
of these along with some other recent advances on graph minors are surveyed.
Research partly supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research, Grant number
16740044, by Sumitomo Foundation, by C & C Foundation and by Inoue Research Award for Young Scientists
Supported in part by the Research Grant P1–0297 and by the CRC program
On leave from: IMFM & FMF, Department of Mathematics, University of Ljubljana, Ljubljana, Slovenia 相似文献
5.
6.
The oscillator strengths of the symmetry-forbidden transitions of the ion [PdCl4]2– were calculated by a method based on the evaluation of the MO of the distorted ion. It is not very likely that the transition to1
B
1g
(a
1g
(d
z
2)b
1g
) state, weak, mainlyz polarized, explains the band that appears as a shoulder observed towards 30 000 cm–1 in the absorption spectrum. If this band is specific of the ion [PdCl4]2–, the assignment to the forbidden charge-transfer transition,1
A
1g
1
B
2g
, is more plausible. 相似文献
7.
Correlation of structure parameters of glasses and related crystals formed in homogeneous or heterogeneous nucleation processes by thermal treatment is discussed on the basis of DTA, TG and EGA measurements in relation to the textural patterns of the materials. For cordierite glass, crystallization of metastable disordered cordierite polymorphs is related to an exothermic heat evolution and simultaneous with a small weight loss (appr. 0.025%). By MS-EGA, evolution of water was determined during the transformation of the metastable melt to a metastable intermediate crystalline phase. Interpretation of the crystallization by comparing the available structure parameters of cordierite glasses and crystals alone is insufficient to explain the role of water in the kinetics of crystallization. Optical and electron microscopy of the primary crystallization phenomena show the metastable solid solution with low quartz-type structure. Interpretation of the crystallization behaviour in terms of conventional theory of nucleation and crystal growth is impossible. 相似文献
8.
Solid solutions NaCl—CdCl2 were studied in an interval of CdCl2 concentrations of 0.05—3 mol.% by Raman spectroscopy. The molecular form Na2CdCl4 decomposes: Na2CdCl4 2Na+ + Cd2+ + 2Cl2
– + 2e–; 2Na + 2e– 2Na0. Free sodium atoms form color centers of crystal (F centers) in the region of jumpwise changing the partition coefficient of CdCl2 in NaCl (K jump). 相似文献
9.
10.