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1.
We say that H has an odd complete minor of order at least l if there are l vertex disjoint trees in H such that every two of them are joined by an edge, and in addition, all the vertices of trees are two-colored in such a way that the edges within the trees are bichromatic, but the edges between trees are monochromatic. Gerards and Seymour conjectured that if a graph has no odd complete minor of order l, then it is (l ? 1)-colorable. This is substantially stronger than the well-known conjecture of Hadwiger. Recently, Geelen et al. proved that there exists a constant c such that any graph with no odd K k -minor is ck√logk-colorable. However, it is not known if there exists an absolute constant c such that any graph with no odd K k -minor is ck-colorable. Motivated by these facts, in this paper, we shall first prove that, for any k, there exists a constant f(k) such that every (496k + 13)-connected graph with at least f(k) vertices has either an odd complete minor of size at least k or a vertex set X of order at most 8k such that G–X is bipartite. Since any bipartite graph does not contain an odd complete minor of size at least three, the second condition is necessary. This is an analogous result of Böhme et al. We also prove that every graph G on n vertices has an odd complete minor of size at least n/2α(G) ? 1, where α(G) denotes the independence number of G. This is an analogous result of Duchet and Meyniel. We obtain a better result for the case α(G)= 3. 相似文献
2.
Benjamin Sambale 《Israel Journal of Mathematics》2018,228(2):513-526
An open conjecture by Harada from 1981 gives an easy characterization of the p-blocks of a finite group in terms of the ordinary character table. Kiyota and Okuyama have shown that the conjecture holds for p-solvable groups. In the present work we extend this result using a criterion on the decomposition matrix. In this way we prove Harada’s Conjecture for several new families of defect groups and for all blocks of sporadic simple groups. In the second part of the paper we present a dual approach to Harada’s Conjecture. 相似文献
3.
It is shown that the difference between Donovans conjecture and the weaker
conjecture bounding Cartan numbers of blocks of finite groups by the defect of
the blocks can be expressed in terms of the relationship between pairs of Galois
conjugate blocks. A consequence is that for principal blocks the two conjectures
are equivalent.Received: 11 August 2003 相似文献
4.
Xiaoyun Lu 《Discrete Mathematics》2011,311(23-24):2711-2715
A well-known conjecture of Barnette states that every 3-connected cubic bipartite planar graph has a Hamiltonian cycle, which is equivalent to the statement that every 3-connected even plane triangulation admits a 2-tree coloring, meaning that the vertices of the graph have a 2-coloring such that each color class induces a tree. In this paper we present a new approach to Barnette’s conjecture by using 2-tree coloring.A Barnette triangulation is a 3-connected even plane triangulation, and a B-graph is a smallest Barnette triangulation without a 2-tree coloring. A configuration is reducible if it cannot be a configuration of a B-graph. We prove that certain configurations are reducible. We also define extendable, non-extendable and compatible graphs; and discuss their connection with Barnette’s conjecture. 相似文献
5.
Luis Dieulefait 《manuscripta mathematica》2012,139(1-2):71-89
In this article we give a proof of Serre’s conjecture for the case of odd level and arbitrary weight. Our proof does not use any modularity lifting theorem in characteristic 2 (moreover, we will not consider at all characteristic 2 representations at any step of our proof). The key tool in the proof is a very general modularity lifting result of Kisin, which is combined with the methods and results of previous articles on Serre’s conjecture by Khare, Wintenberger, and the author, and modularity results of Schoof for abelian varieties of small conductor. Assuming GRH, infinitely many cases of even level will also be proved. 相似文献
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7.
Yong-Gao Chen 《Comptes Rendus Mathematique》2012,350(21-22):933-935
8.
The Turán number is the maximum number of edges in any -vertex graph that does not contain a subgraph isomorphic to . A wheel is a graph on vertices obtained from a by adding one vertex and making adjacent to all vertices of the . We obtain two exact values for small wheels: Given that is already known, this paper completes the spectrum for all wheels up to 7 vertices. In addition, we present the construction which gives us the lower bound in general case. 相似文献
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10.
Venkatramani Lakshmibai Komaranapuram N. Raghavan Parameswaran Sankaran 《Central European Journal of Mathematics》2009,7(2):214-223
It is shown that the proof by Mehta and Parameswaran of Wahl’s conjecture for Grassmannians in positive odd characteristics
also works for symplectic and orthogonal Grassmannians.
相似文献
11.
JinHua Wang 《中国科学 数学(英文版)》2012,55(6):1215-1220
For a(1+3)-dimensional Lorentzian manifold(M,g),the general form of solutions of the Einstein field equations takes that of type I,II,or III.For type I,there is a known result in Gu(2007).In this paper,we try to find the necessary and sufficient conditions for the Lorentzian metric to take the form of types II and III,and we show how to construct the new coordinate system. 相似文献
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14.
We prove a positive combinatorial formula for the Schur expansion of LLT polynomials indexed by a 3-tuple of skew shapes. This verifies a conjecture of Haglund (Proc Natl Acad Sci USA 101(46):16127–16131, 2004). The proof requires expressing a noncommutative Schur function as a positive sum of monomials in Lam’s (Eur J Combin 29(1):343–359, 2008) algebra of ribbon Schur operators. Combining this result with the expression of Haglund et al. (J Am Math Soc 18(3):735–761, 2005) for transformed Macdonald polynomials in terms of LLT polynomials then yields a positive combinatorial rule for transformed Macdonald polynomials indexed by a shape with 3 columns. 相似文献
15.
Sheng Yang 《Archiv der Mathematik》2011,96(5):401-408
Let G be a finite group, and let B be a p-block of G with defect group D. Let k
0(B) denote the number of ordinary irreducible characters of height 0 in B. In 1984 Olsson proposed a conjecture:
k0(B)\leqq |D:D¢|{k_{0}(B)\leqq |D:D'|}. In this paper, we will verify Olsson’s conjecture in the case that D is metacyclic and p is odd. 相似文献
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17.
On Hua-Tuan’s conjecture 总被引:2,自引:0,他引:2
Let G be a finite group and |G| = pn, p be a prime. For 0 m n, sm(G) denotes the number of subgroups of of order pm of G. Loo-Keng Hua and Hsio-Fu Tuan have ever conjectured: for an arbitrary finite p-group G, if p > 2, then sm(G) ≡ 1, 1 + p, 1 + p + p2 or 1 + p + 2p2 (mod p3). In this paper, we investigate the conjecture, and give some p-groups in which the conjecture holds and some examples in which the conjecture does not hold. 相似文献
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19.
Marcin Pilipczuk Michał Pilipczuk Riste Škrekovski 《Discrete Applied Mathematics》2012,160(16-17):2484-2490
The well-known conjecture of Vizing on the domination number of Cartesian product graphs claims that for any two graphs and , . We disprove its variations on independent domination number and Barcalkin–German number, i.e. Conjectures 9.6 and 9.2 from the recent survey Bre?ar et al. (2012) [4]. We also give some extensions of the double-projection argument of Clark and Suen (2000) [8], showing that their result can be improved in the case of bounded-degree graphs. Similarly, for rainbow domination number we show for every that , which is closely related to Question 9.9 from the same survey. We also prove that the minimum possible counterexample to Vizing’s conjecture cannot have two neighboring vertices of degree two. 相似文献