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1.
Carsten Thomassen 《Combinatorica》2007,27(2):241-243
We prove that, for each fixed real number c > 0, the pentagon-free graphs of minimum degree at least cn (where n is the number of vertices) have bounded chromatic number. This problem was raised by Erdős and Simonovits in 1973. A similar
result holds for any other fixed odd cycle, except the triangle for which there is no such result for c<1/3. 相似文献
2.
Circular Chromatic Number and
Mycielski Graphs 总被引:7,自引:0,他引:7
As a natural generalization of graph coloring, Vince
introduced the star chromatic number of a graph
G and denoted it by
*(G). Later, Zhu called it circular
chromatic number and denoted it by c(G). Let (G)
be the chromatic number of G.
In this paper, it is shown that if the complement of
G is non-hamiltonian, then
c(G)=(G). Denote by M(G)
the Mycielski graph of G.
Recursively define Mm(G)=M(Mm–1(G)). It was conjectured that if
mn–2, then c(Mm(Kn))=(Mm(Kn)). Suppose that
G is a graph on n vertices.
We prove that if
, then
c(M(G))=(M(G)). Let S be the set of vertices of degree
n–1 in
G. It is proved that if
|S| 3, then
c(M(G))=(M(G)), and if |S| 5, then c(M2(G))=(M2(G)), which implies the known results of
Chang, Huang, and Zhu that if n3, c(M(Kn))=(M(Kn)), and if
n5, then
c(M2(Kn))=(M2(Kn)).* Research supported by Grants from National Science
Foundation of China and Chinese Academy of
Sciences. 相似文献
3.
A circulation in a digraph is a spanning subgraph with indegree equal to outdegree at each vertex. Alon and Tarsi proved that a graph is d-choosable when it has an orientation that has no vertex of outdegree at least d and has the property that the numbers of circulations with even-sized and odd-sized edge sets differ. We generalize this to k-uniform hypergraphs for prime k. We use a hypergraph polynomial and a notion of hypergraph orientation defined by choosing a source vertex from each edge.* Project sponsored by the National Security Agency under Grant Number MDA904-03-1-0037. The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation herein. 相似文献
4.
We prove that, for each xed real number
c > 1/3, the triangle-free
graphs of minimum degree at least cn (where n is the number of vertices) have
bounded chromatic number. This problem was raised by Erds and
Simonovits in 1973 who pointed out that there is no such result
for c < 1/3. 相似文献
5.
We present several partial results, variants, and
consistency results concerning the following (as yet unsolved)
conjecture. If X is a graph
on the ground set V with
then
X has an edge coloring
F with
colors such that if
V is decomposed into
parts then there is one
in which F assumes all
values.Due to some unfortunate misunderstandings, this
paper appeared much later than we expected.* Research partially supported by NSF grants
DMS-9704477 and DMS-0072560. Research partially supported by Hungarian National
Research Grant T 032455. 相似文献
6.
7.
Claude Tardif 《Combinatorica》2005,25(5):625-632
We prove that the identity
holds for all directed graphs G and H. Similar bounds for the usual chromatic number seem to be much harder to obtain: It is still not known whether there exists
a number n such that χ(G×H) ≥ 4 for all directed graphs G, H with χ(G) ≥ χ(H) ≥ n. In fact, we prove that for every integer n ≥ 4, there exist directed graphs Gn, Hn such that χ(Gn) = n, χ(Hn) = 4 and χ(Gn×Hn) = 3. 相似文献
8.
A sufficient condition for graphs with circular flow index less than 4 is found in this paper. In particular, we give a simple
proof of a result obtained by Galluccio and Goddyn (Combinatorica, 2002), and obtain a larger family of such graphs.
* Partially supported by the National Security Agency under Grants MDA904-01-1-0022. 相似文献
9.
This article intends to study some functors from the category of graphs to itself such that, for any graph G, the circular chromatic number of is determined by that of G. In this regard, we investigate some coloring properties of graph powers. We show that provided that . As a consequence, we show that if , then . In particular, and has no subgraph with circular chromatic number equal to . This provides a negative answer to a question asked in (X. Zhu, Discrete Math, 229(1–3) (2001), 371–410). Moreover, we investigate the nth multichromatic number of subdivision graphs. Also, we present an upper bound for the fractional chromatic number of subdivision graphs. Precisely, we show that . 相似文献
10.
《Quaestiones Mathematicae》2013,36(7):953-975
AbstractEvery partial colouring of a Hamming graph is uniquely related to a partial Latin hyper-rectangle. In this paper we introduce the Θ-stabilized (a, b)-colouring game for Hamming graphs, a variant of the (a, b)-colouring game so that each move must respect a given autotopism Θ of the resulting partial Latin hyperrectangle. We examine the complexity of this variant by means of its chromatic number. We focus in particular on the bi-dimensional case, for which the game is played on the Cartesian product of two complete graphs, and also on the hypercube case. 相似文献
11.
The local chromatic number of a graph was introduced in [14]. It is in between the chromatic and fractional chromatic numbers.
This motivates the study of the local chromatic number of graphs for which these quantities are far apart. Such graphs include
Kneser graphs, their vertex color-critical subgraphs, the Schrijver (or stable Kneser) graphs; Mycielski graphs, and their
generalizations; and Borsuk graphs. We give more or less tight bounds for the local chromatic number of many of these graphs.
We use an old topological result of Ky Fan [17] which generalizes the Borsuk–Ulam theorem. It implies the existence of a multicolored
copy of the complete bipartite graph K⌈t/2⌉,⌊t/2⌋ in every proper coloring of many graphs whose chromatic number t is determined via a topological argument. (This was in particular noted for Kneser graphs by Ky Fan [18].) This yields a
lower bound of ⌈t/2⌉ + 1 for the local chromatic number of these graphs. We show this bound to be tight or almost tight in many cases.
As another consequence of the above we prove that the graphs considered here have equal circular and ordinary chromatic numbers
if the latter is even. This partially proves a conjecture of Johnson, Holroyd, and Stahl and was independently attained by
F. Meunier [42]. We also show that odd chromatic Schrijver graphs behave differently, their circular chromatic number can
be arbitrarily close to the other extreme.
* Research partially supported by the Hungarian Foundation for Scientific Research Grant (OTKA) Nos. T037846, T046376, AT048826,
and NK62321.
† Research partially supported by the NSERC grant 611470 and the Hungarian Foundation for Scientific Research Grant (OTKA)
Nos. T037846, T046234, AT048826, and NK62321. 相似文献
12.
. The proof is probabilistic.
Received: November 26, 1996 相似文献
13.
《Quaestiones Mathematicae》2013,36(2):217-232
AbstractIn this paper, general results on the toughness of a graph are considered. Firstly the link between toughness and connectivity is explored and then results linking toughness and the parameters binding number and integrity are given. Further, the toughness of product graphs is discussed including general results for the lexicographic product. The paper concludes with some observations on toughness and hamiltoni-city. 相似文献
14.
A set S of integers is called a cycle set on {1, 2, . . .,n} if there exists a graph G on n vertices such that the set of lengths of cycles in G is S. Erds conjectured that the number of cycle sets on {1, 2, . . .,n} is o(2
n
). In this paper, we verify this conjecture by proving that there exists an absolute constant c 0.1 such that the number of cycle sets on {1, 2, . . .,n} is
. 相似文献
15.
Graph Connectivity After Path
Removal 总被引:1,自引:0,他引:1
Let G be a graph and
u, v be two distinct vertices of
G. A u—v path P is called nonseparating if
G—V(P) is connected. The purpose of this
paper is to study the number of nonseparating
u—v path for two arbitrary
vertices u and
v of a given graph. For a
positive integer k, we will
show that there is a minimum integer (k) so that if G is an (k)-connected graph and
u and v are two arbitrary vertices in
G, then there exist
k vertex disjoint paths
P
1[u,v], P
2[u,v], . . ., P
k
[u,v], such that G—V (P
i
[u,v]) is connected for every
i (i = 1, 2, ..., k). In fact, we will prove that
(k) 22k+2. It is known that (1) = 3.. A
result of Tutte showed that (2) = 3. We show that (3) = 6. In
addition, we prove that if G
is a 5-connected graph, then for every pair of vertices
u and v there exists a path
P[u, v] such that G—V(P[u,
v]) is 2-connected.* Supported by NSF grant No.
DMS-0070059 Supported by ONR grant
N00014-97-1-0499 Supported by NSF grant No. 9531824 相似文献
16.
Let T be a fixed tournament on k vertices. Let D(n,T ) denote the maximum number of orientations of an n-vertex graph that have no copy of T. We prove that
for all sufficiently (very) large n, where tk−1(n) is the maximum possible number of edges of a graphon n vertices with no Kk, (determined by Turán’s Theorem). The proof is based on a directed version of Szemerédi’s regularity lemma together with
some additional ideas and tools from Extremal Graph Theory, and provides an example of a precise result proved by applying
this lemma. For the two possible tournaments with three vertices we obtain separate proofs that avoid the use of the regularity
lemma and therefore show that in these cases
already holds for (relatively) small values of n.
* Research supported in part by a USA Israeli BSF grant, by a grant from the Israel Science Foundation and by the Hermann
Minkowski Minerva Center for Geometry at Tel Aviv University. 相似文献
17.
In this paper, we derive some necessary spectral conditions for the existence of graph homomorphisms in which we also consider some parameters related to the corresponding eigenspaces such as nodal domains. In this approach, we consider the combinatorial Laplacian and co-Laplacian as well as the adjacency matrix. Also, we present some applications in graph decompositions where we prove a general version of Fisher’s inequality for G-designs. 相似文献
18.
A congruence (p is a prime) is said to be strong homogeneous if it has the form
19.
《Quaestiones Mathematicae》2013,36(4):537-548
AbstractFor a set F of graphs and a natural number k, an (F, k)-colouring of a graph G is a proper colouring of V (G) such that no subgraph of G isomorphic to an element of F is coloured with at most k colours. Equivalently, if P is the class of all graphs that do not contain an element of F as a subgraph, a χP,k colouring of G is a proper colouring such that the union of at most k colour classes induces a graph in P. The smallest number of colours in such a colouring of G, if it exists, is denoted by χP,k (G). We give some general results on χP,k-colourings and investigate values of χP,k (G) for some choices of P and classes of graphs G. 相似文献
20.
We (1) determine the number of Latin rectangles with 11 columns and each possible number of rows, including the Latin squares
of order 11, (2) answer some questions of Alter by showing that the number of reduced Latin squares of order n is divisible by f! where f is a particular integer close to
(3) provide a formula for the number of Latin squares in terms of permanents of (+1, −1)-matrices, (4) find the extremal
values for the number of 1-factorisations of k-regular bipartite graphs on 2n vertices whenever 1 ≤ k ≤ n ≤ 11, (5) show that the proportion of Latin squares with a non-trivial symmetry group tends quickly to zero as the order
increases.
Received September 3, 2004 相似文献