Báo cáo toán hoc:" Defective choosability of graphs without small minors" docx

... electronic journal of combinatorics 16 (2009), #R92 2 Defective choosability of graphs without small minors Rupert G. Woo d and Douglas R. Woodall School of Mathematical Sciences, University of Nottingham, Nottingham ... and S. Hedetniemi, Graphs with forbidden subgraphs. J. Combin. Theory Ser. B 10 (1971) 12–41. [3] G. A. Dirac, A property of 4-chromatic graphs and so...

Ngày tải lên: 08/08/2014, 01:20

... blue clique. Graphs of maximum order without this property will be called P(r, b)−critical. The goal of this paper is to prove the following Theorem. If P is the class of all graphs of chromatic ... lower bounds. Examples of critical P (r, b) graphs are disjoint unions of b − 1 copies of of complete k-partite graphs in which every part has r − 1 vertices. Other critica...

Ngày tải lên: 07/08/2014, 06:20

... graphs with a somewhat lower maximum degree, but still with ∆(G)=Ω( √ n), always have a k-OOC.Thisisshown in Theorem 1.5: Proof of Theorem 1.5: Consider the set V of the vertices of G,asasetofn ... orthogonal chromatic number of complete partite graphs canbecomputedexactly. Proof of Theorem 1.3: Let S 1 , ,S t denote the vertex classes of G,where|S i | = s i , and the sizes o...

Ngày tải lên: 07/08/2014, 06:20

... parts are all of size one. Alternately, the complement of such a graph either has no edges or consists of the the disjoint union of three or fewer complete graphs. Proof. The complement of K − 5 is ... similar proofs for complexes of dimension 2 and higher. We conjecture that the answer is “yes,” but the complexity of the proof would be astronomical, even for dimension 2. Hidd...

Ngày tải lên: 07/08/2014, 06:20

... K 5 -minor- free. Hence, the class of (maximal) planar graphs is contained in the class of maximal K 3,3 -minor-free graphs and we can use the number of planar graphs and the number of triangulations as lower ... omit the proof as it follows closely the lines of proof of Lemma 5 in [7]. Lemma 3.5. Let W (x, z) = z(1+U(x, z)). The generating function B(x, y) of 2-connected...

Ngày tải lên: 07/08/2014, 21:20

... graphs remains unknown. We prove a lower bound of 1 4 n 1/3 and conjecture that the truth is Θ(n 1/2 ). For line graphs of trees, the minimum is Θ(n 1/2 ). For several other classes of line graphs, ... families of sparse graphs with a linear number of edges, Theorem 4.1 gives lower bounds that are multiples of n 1/2 instead of the previous n 1/3 . With Proposition 3.4, we...

Ngày tải lên: 07/08/2014, 21:21

... element of the stabilizer of S. This clearly requires at least ⌈ n−1 k+1 ⌉ vertices, so fix(Inf k (K n )) = ⌈ n−1 k+1 ⌉. It seems likely that the proof of Theorem 1 6 could extend to inflatio ns of graphs ... of G if stab(S) is trivia l. In this case we say that S fixes G. The fixing number fix(G) of a graph G is the smallest cardinality of a fixing set of G [3, 5, 9]. Equivalentl...

Ngày tải lên: 07/08/2014, 21:21

... The set of partial latin squares of order n is denoted by PLS(n), and the set of latin squares of order n by LS(n). We say that P 1 ∈ PLS(n) contains P 2 ∈ PLS(n) if every filled cell of P 2 agrees ... Cayley table of G is invariant with the enumeration chosen for G, so to complete the proof of the theorem it suffices to exhibit a 2-partition of L. We do this by first arguing that...

Ngày tải lên: 07/08/2014, 06:23

... value of f (p, G) taken over all K r+1 -free graphs G of order n, Caro and Yuster stated that, for every p ≥ 1, φ (r, p, n)=f (p, T r (n)) , (1) ∗ Department of Mathematical Sciences, University of ... and sufficiently large n, his proof is valid, since it is independent of the exact value of φ (r, p, n) . Our proof is close to Pikhurko’s, and is given only for the sake of compl...

Ngày tải lên: 07/08/2014, 08:20

... partially ordered set of all subsets of [r]. Proof. By Lemma 3, each P ∈P k (u, v) is the union of some (possibly none) of the paths P i together with internally disjoint subpaths of P ∗ . Therefore ... new series of dense graphs of high girth, Bull. Amer. Math. Soc. 32 (1995), 73–79. [9] F. Lazebnik, V. A. Ustimenko and A. J. Woldar, Properties of Certain Families of 2k–C...

Ngày tải lên: 07/08/2014, 08:22