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1.
C. Thomassen proposed a conjecture: Let G be a k‐connected graph with the stability number α ≥ k, then G has a cycle C containing k independent vertices and all their neighbors. In this paper, we will obtain the following result: Let G be a k‐connected graph with stability number α = k + 3 and C any longest cycle of G, then C contains k independent vertices and all their neighbors. This solves Thomassen's conjecture for the case α = k + 3. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 168–180, 2001  相似文献   

2.
Enomoto 7 conjectured that if the minimum degree of a graph G of order n ≥ 4k ? 1 is at least the integer , then for any k vertices, G contains k vertex‐disjoint cycles each of which contains one of the k specified vertices. We confirm the conjecture for n ≥ ck2 where c is a constant. Furthermore, we show that under the same condition the cycles can be chosen so that each has length at most six. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 276–296, 2003  相似文献   

3.
Let R(G) denote the minimum integer N such that for every bicoloring of the edges of KN, at least one of the monochromatic subgraphs contains G as a subgraph. We show that for every positive integer d and each γ,0 < γ < 1, there exists k = k(d,γ) such that for every bipartite graph G = (W,U;E) with the maximum degree of vertices in W at most d and , . This answers a question of Trotter. We give also a weaker bound on the Ramsey numbers of graphs whose set of vertices of degree at least d + 1 is independent. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 198–204, 2001  相似文献   

4.
Daniel Finkel   《Discrete Mathematics》2008,308(22):5265-5268
Hajnal and Corrádi proved that any simple graph on at least 3k vertices with minimal degree at least 2k contains k independent cycles. We prove the analogous result for chorded cycles. Let G be a simple graph with |V(G)|4k and minimal degree δ(G)3k. Then G contains k independent chorded cycles. This result is sharp.  相似文献   

5.
Erdoes and Soes conjectured in 1963 that every graph G on n vertices with edge number e(G) 〉 1/2(k - 1)n contains every tree T with k edges as a subgraph. In this paper, we consider a variation of the above conjecture, that is, for n 〉 9/ 2k^2 + 37/2+ 14 and every graph G on n vertices with e(G) 〉 1/2 (k- 1)n, we prove that there exists a graph G' on n vertices having the same degree sequence as G and containing every tree T with k edges as a subgraph.  相似文献   

6.
Let denote the set of graphs with each vertex of degree at least r and at most s, v(G) the number of vertices, and τk (G) the maximum number of disjoint k‐edge trees in G. In this paper we show that
  • (a1) if G ∈ and s ≥ 4, then τ2(G) ≥ v(G)/(s + 1),
  • (a2) if G ∈ and G has no 5‐vertex components, then τ2(G) ≥ v(G)4,
  • (a3) if G ∈ and G has no k‐vertex component, where k ≥ 2 and s ≥ 3, then τk(G) ≥ (v(G) ‐k)/(skk + 1), and
  • (a4) the above bounds are attained for infinitely many connected graphs.
Our proofs provide polynomial time algorithms for finding the corresponding packings in a graph. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 306–324, 2007  相似文献   

7.
It is shown that if G is a graph of order n with minimum degree δ(G), then for any set of k specified vertices {v1,v2,…,vk} ? V(G), there is a 2‐factor of G with precisely k cycles {C1,C2,…,Ck} such that viV(Ci) for (1 ≤ ik) if or 3k + 1 ≤ n ≤ 4k, or 4kn ≤ 6k ? 3,δ(G) ≥ 3k ? 1 or n ≥ 6k ? 3, . Examples are described that indicate this result is sharp. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 188–198, 2003  相似文献   

8.
A graph G is k‐choosable if its vertices can be colored from any lists L(ν) of colors with |L(ν)| ≥ k for all ν ∈ V(G). A graph G is said to be (k,?)‐choosable if its vertices can be colored from any lists L(ν) with |L(ν)| ≥k, for all ν∈ V(G), and with . For each 3 ≤ k ≤ ?, we construct a graph G that is (k,?)‐choosable but not (k,? + 1)‐choosable. On the other hand, it is proven that each (k,2k ? 1)‐choosable graph G is O(k · ln k · 24k)‐choosable. © 2005 Wiley Periodicals, Inc. J Graph Theory  相似文献   

9.
Let G be a graph of order n ≥ 5k + 2, where k is a positive integer. Suppose that the minimum degree of G is at least ?(n + k)/2?. We show that G contains k pentagons and a path such that they are vertex‐disjoint and cover all the vertices of G. Moreover, if n ≥ 5k + 7, then G contains k + 1 vertex‐disjoint cycles covering all the vertices of G such that k of them are pentagons. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 194–208, 2007  相似文献   

10.
It is proved that every graph G with ‖G‖ ≥ 2|G| − 5, |G| ≥ 6, and girth at least 5, except the Petersen graph, contains a subdivision of K, the complete graph on five vertices minus one edge. © 1999 John Wiley & Sons, Inc, J. Graph Theory 30: 261–276, 1999  相似文献   

11.
Given lists of available colors assigned to the vertices of a graph G, a list coloring is a proper coloring of G such that the color on each vertex is chosen from its list. If the lists all have size k, then a list coloring is equitable if each color appears on at most vertices. A graph is equitably k-choosable if such a coloring exists whenever the lists all have size k. We prove that G is equitably k-choosable when unless G contains or k is odd and . For forests, the threshold improves to . If G is a 2-degenerate graph (given k ≥ 5) or a connected interval graph (other than ), then G is equitably k-choosable when . © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 166–177, 2003  相似文献   

12.
Let G be a K1,r ‐free graph (r ≥ 3) on n vertices. We prove that, for any induced path or induced cycle on k vertices in G (k ≥ 2r − 1 or k ≥ 2r, respectively), the degree sum of its vertices is at most (2r − 2)(n − α) where α is the independence number of G. As a corollary we obtain an upper bound on the length of a longest induced path and a longest induced cycle in a K1,r ‐free graph. Stronger bounds are given in the special case of claw‐free graphs (i.e., r = 3). Sharpness examples are also presented. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 131–143, 2001  相似文献   

13.
Let r(k) denote the least integer n-such that for any graph G on n vertices either G or its complement G contains a complete graph Kk on k vertices. in this paper, we prove the following lower bound for the Ramsey number r(k) by explicit construction: r(k) ≥ exp (c(Log k)4/3[(log log k)1/3] for some constant c> 0.  相似文献   

14.
Let k and n be two integers such that k ≥ 0 and n ≥ 3(k + 1). Let G be a graph of order n with minimum degree at least ?(n + k)/2?. Then G contains k + 1 independent cycles covering all the vertices of G such that k of them are triangles. © 1995, John Wiley & Sons, Inc.  相似文献   

15.
Dirac proved that a graph G is hamiltonian if the minimum degree , where n is the order of G. Let G be a graph and . The neighborhood of A is for some . For any positive integer k, we show that every (2k ? 1)‐connected graph of order n ≥ 16k3 is hamiltonian if |N(A)| ≥ n/2 for every independent vertex set A of k vertices. The result contains a few known results as special cases. The case of k = 1 is the classic result of Dirac when n is large and the case of k = 2 is a result of Broersma, Van den Heuvel, and Veldman when n is large. For general k, this result improves a result of Chen and Liu. The lower bound 2k ? 1 on connectivity is best possible in general while the lower bound 16k3 for n is conjectured to be unnecessary. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 83–100, 2006  相似文献   

16.
A homomorphism from an oriented graph G to an oriented graph H is a mapping from the set of vertices of G to the set of vertices of H such that is an arc in H whenever is an arc in G. The oriented chromatic index of an oriented graph G is the minimum number of vertices in an oriented graph H such that there exists a homomorphism from the line digraph LD(G) of G to H (the line digraph LD(G) of G is given by V(LD(G)) = A(G) and whenever and ). We give upper bounds for the oriented chromatic index of graphs with bounded acyclic chromatic number, of planar graphs and of graphs with bounded degree. We also consider lower and upper bounds of oriented chromatic number in terms of oriented chromatic index. We finally prove that the problem of deciding whether an oriented graph has oriented chromatic index at most k is polynomial time solvable if k ≤ 3 and is NP‐complete if k ≥ 4. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 313–332, 2008  相似文献   

17.
A graph G of order n satisfies the neighborhood condition NCk > s if any collection of k independent vertices is collectively adjacent to more than s vertices. Given a family H of graphs, the decomposition class β(H) is the family of graphs B with the property that for some HH of chromatic number d, H contains B as an induced subgraph and l?V(H) ? V(B)? is (d ? 2) colorable. Let H be a family of d-chromatic graphs, B an element of β(H) such that B contains an s-matching as an induced subgraph. Thus the cardinality of one of the partite sets of B is s + r for some integer r ≥ 0. We show that if t is a fixed positive integer, G is a graph of sufficiently large order n that satisfies the neighborhood condition then G contains B + K(d - 2; t) as a subgraph.  相似文献   

18.
Ohba has conjectured that if G is a k-chromatic graph with at most 2k+1 vertices, then the list chromatic number or choosability of G is equal to its chromatic number χ(G), which is k. It is known that this holds if G has independence number at most three. It is proved here that it holds if G has independence number at most five. In particular, and equivalently, it holds if G is a complete k-partite graph and each part has at most five vertices.  相似文献   

19.
A k-cube (or “a unit cube in k dimensions”) is defined as the Cartesian product where R i (for 1 ≤ i ≤ k) is an interval of the form [a i , a i  + 1] on the real line. The k-cube representation of a graph G is a mapping of the vertices of G to k-cubes such that the k-cubes corresponding to two vertices in G have a non-empty intersection if and only if the vertices are adjacent. The cubicity of a graph G, denoted as cub(G), is defined as the minimum dimension k such that G has a k-cube representation. An interval graph is a graph that can be represented as the intersection of intervals on the real line - i.e., the vertices of an interval graph can be mapped to intervals on the real line such that two vertices are adjacent if and only if their corresponding intervals overlap. We show that for any interval graph G with maximum degree Δ, . This upper bound is shown to be tight up to an additive constant of 4 by demonstrating interval graphs for which cubicity is equal to .  相似文献   

20.
An mcovering of a graph G is a spanning subgraph of G with maximum degree at most m. In this paper, we shall show that every 3‐connected graph on a surface with Euler genus k ≥ 2 with sufficiently large representativity has a 2‐connected 7‐covering with at most 6k ? 12 vertices of degree 7. We also construct, for every surface F2 with Euler genus k ≥ 2, a 3‐connected graph G on F2 with arbitrarily large representativity each of whose 2‐connected 7‐coverings contains at least 6k ? 12 vertices of degree 7. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 26–36, 2003  相似文献   

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