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1.
Jianguo Qian 《Graphs and Combinatorics》2006,22(3):391-398
A graph G is called induced matching extendable (shortly, IM-extendable) if every induced matching of G is included in a perfect matching of G. A graph G is called strongly IM-extendable if every spanning supergraph of G is IM-extendable. The k-th power of a graph G, denoted by Gk, is the graph with vertex set V(G) in which two vertices are adjacent if and only if the distance between them in G is at most k.
We obtain the following two results which give positive answers to two conjectures of Yuan.
Result 1. If a connected graph G with |V(G)| even is locally connected, then G2 is strongly IM-extendable.
Result 2. If G is a 2-connected graph with |V(G)| even, then G3 is strongly IM-extendable.
Research Supported by NSFC Fund 10371102. 相似文献
2.
A graph G is induced matching extendable, shortly IM-extendable, if every induced matching of G is included in a perfect matching of G. For a nonnegative integer k, a graph G is called a k-edge-deletable IM-extendable graph, if, for every F⊆E(G) with |F|=k, G−F is IM-extendable. In this paper, we characterize the k-edge-deletable IM-extendable graphs with minimum number of edges. We show that, for a positive integer k, if G is ak-edge-deletable IM-extendable graph on 2n vertices, then |E(G)|≥(k+2)n; furthermore, the equality holds if and only if either G≅Kk+2,k+2, or k=4r−2 for some integer r≥3 and G≅C5[N2r], where N2r is the empty graph on 2r vertices and C5[N2r] is the graph obtained from C5 by replacing each vertex with a graph isomorphic to N2r. 相似文献
3.
We present a new condition on the degree sums of a graph that implies the existence of a long cycle. Let c(G) denote the length of a longest cycle in the graph G and let m be any positive integer. Suppose G is a 2-connected graph with vertices x1,…,xn and edge set E that satisfies the property that, for any two integers j and k with j < k, xjxk ? E, d(xi) ? j and d(xk) ? K - 1, we have (1) d(xi) + d(xk ? m if j + k ? n and (2) if j + k < n, either m ? n or d(xj) + d(xk) ? min(K + 1,m). Then c(G) ? min(m, n). This result unifies previous results of J.C. Bermond and M. Las Vergnas, respectively. 相似文献
4.
Say that graph G is partitionable if there exist integers α?2, ω? 2, such that |V(G)| ≡ αω + 1 and for every υ?V(G) there exist partitions of V(G)\ υ into stable sets of size α and into eliques of size ω. An immediate consequence of Lovász' characterization of perfect graphs is that every minimal imperfect graph G is partitionable with α ≡ α (G) andω ≡ ω(G).Padberg has shown that in every minimal imperfect graph G the cliques and stable sets of maximum size satisfy a series of conditions that reflect extraordinary symmetry G. Among these conditions are: the number of cliques of size ω(G) is exactly |V(G)|; the number of stable sets of size α(G) is exactly |V(G)|: every vertex of G is contained in exactly ω(G) cliques of size ω(G) and α(G) stable sets of size α(G): for every clique Q (respectively, stable set S) of maximum size there is a unique stable set S (clique O) of maximum size such that Q∩S ≡ Ø.Let Cnk denote the graph whose vertices can be enumerated as υ1,…,υn in such a way that υ1 and υ1 are adjacent in G if and only if i and j differ by at most k, modulo n. Chvátal has shown that Berge's Strong Perfect graph Conjecture is equivalent to the conjecture that if G is minimal imperfect with α(G) ≡ αandω(G) ≡ ω, then G has a spanning subgraph isomorphic to Cαω+1ω. Padberg's conditions are sufficiently restrictive to suggest the possibility of establishing the Strong Perfect Graph Conjecture by proving that any graph G satisfying these conditions must contain a spanning subgraph isomorphic to Cαω+1ω, whereα(G) ≡ αandω(G) ≡ ω. It is shown here, using only elementary linear algebra, that all partitionable graphs satisfy Padberg's conditions, as well as additional properties of the same spirit. Then examples are provided of partitionable graphs which contain no spanning subgraph isomorphic to Cαω+1ω, whereα(G) ≡ α and ω(G) ≡ ω. 相似文献
5.
6.
For a graph G, let σ2(G) denote the minimum degree sum of a pair of nonadjacent vertices. We conjecture that if |V(G)| = n = Σki = 1 ai and σ2(G) ≥ n + k − 1, then for any k vertices v1, v2,…, vk in G, there exist vertex‐disjoint paths P1, P2,…, Pk such that |V(Pi)| = ai and vi is an endvertex of Pi for 1 ≤ i ≤ k. In this paper, we verify the conjecture for the cases where almost all ai ≤ 5, and the cases where k ≤ 3. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 163–169, 2000 相似文献
7.
Hong Lin Xiao-feng Guo 《应用数学学报(英文版)》2007,23(1):155-160
Let φ(G),κ(G),α(G),χ(G),cl(G),diam(G)denote the number of perfect matchings,connectivity,independence number,chromatic number,clique number and diameter of a graph G,respectively.In this note,by constructing some extremal graphs,the following extremal problems are solved:1.max{φ(G):|V(G)|=2n,κ(G)≤k}=k[(2n-3)!!],2.max{φ(G):|V(G)|=2n,α(G)≥k}=[multiply from i=0 to k-1(2n-k-i)[(2n-2k-1)!!],3.max{φ(G):|V(G)|=2n,χ(G)≤k}=φ(T_(k,2n))T_(k,2n)is the Turán graph,that is a complete k-partite graphon 2n vertices in which all parts are as equal in size as possible,4.max{φ(G):|V(G)|=2n,cl(G)=2}=n1,5.max{φ(G):|V(G)|=2n,diam(G)≥2}=(2n-2)(2n-3)[(2n-5)!!],max{φ(G):|V(G)|=2n,diam(G)≥3}=(n-1)~2[(2n-5)!!]. 相似文献
8.
Hong Wang 《Journal of Graph Theory》1997,26(2):105-109
We propose a conjecture: for each integer k ≥ 2, there exists N(k) such that if G is a graph of order n ≥ N(k) and d(x) + d(y) ≥ n + 2k - 2 for each pair of non-adjacent vertices x and y of G, then for any k independent edges e1, …, ek of G, there exist k vertex-disjoint cycles C1, …, Ck in G such that ei ∈ E(Ci) for all i ∈ {1, …, k} and V(C1 ∪ ···∪ Ck) = V(G). If this conjecture is true, the condition on the degrees of G is sharp. We prove this conjecture for the case k = 2 in the paper. © 1997 John Wiley & Sons, Inc. J Graph Theory 26: 105–109, 1997 相似文献
9.
In 1990 G. T. Chen proved that if G is a 2-connected graph of order n and 2|N(x) ∪ N(y)| + d(x) + d(y) ≥ 2n − 1 for each pair of nonadjacent vertices x, y ∈ V (G), then G is Hamiltonian. In this paper we prove that if G is a 2-connected graph of order n and 2|N(x) ∪ N(y)| + d(x)+d(y) ≥ 2n−1 for each pair of nonadjacent vertices x, y ∈ V (G) such that d(x, y) = 2, then G is Hamiltonian. 相似文献
10.
Jarmila Chvátalová 《Discrete Mathematics》1975,11(3):249-253
If G is a graph with p vertices and at least one edge, we set φ (G) = m n max |f(u) ? f(v)|, where the maximum is taken over all edges uv and the minimum over all one-to-one mappings f : V(G) → {1, 2, …, p}: V(G) denotes the set of vertices of G.Pn will denote a path of length n whose vertices are integers 1, 2, …, n with i adjacent to j if and only if |i ? j| = 1. Pm × Pn will denote a graph whose vertices are elements of {1, 2, …, m} × {1, 2, …, n} and in which (i, j), (r, s) are adjacent whenever either i = r and |j ? s| = 1 or j = s and |i ? r| = 1.Theorem.If max(m, n) ? 2, thenφ(Pm × Pn) = min(m, n). 相似文献
11.
Xin Min Hou 《数学学报(英文版)》2012,28(11):2161-2168
A graph G satisfies the Ore-condition if d(x) + d(y) ≥ | V (G) | for any xy ■ E(G). Luo et al. [European J. Combin., 2008] characterized the simple Z3-connected graphs satisfying the Ore-condition. In this paper, we characterize the simple Z3-connected graphs G satisfying d(x) + d(y) ≥ | V (G) |-1 for any xy ■ E(G), which improves the results of Luo et al. 相似文献
12.
Suppose G is a graph and T is a set of non-negative integers that contains 0. A T-coloring of G is an assignment of a non-negative integer f(x) to each vertex x of G such that |f(x)−f(y)|∉T whenever xy∈E(G). The edge span of a T-coloring−f is the maximum value of |f(x) f(y)| over all edges xy, and the T-edge span of a graph G is the minimum value of the edge span of a T-coloring of G. This paper studies the T-edge span of the dth power C
d
n of the n-cycle C
n for T={0, 1, 2, …, k−1}. In particular, we find the exact value of the T-edge span of C
n
d for n≡0 or (mod d+1), and lower and upper bounds for other cases.
Received: May 13, 1996 Revised: December 8, 1997 相似文献
13.
One of the most fundamental results concerning paths in graphs is due to Ore: In a graph G, if deg x + deg y ≧ |V(G)| + 1 for all pairs of nonadjacent vertices x, y ? V(G), then G is hamiltonian-connected. We generalize this result using set degrees. That is, for S ? V(G), let deg S = |x?S N(x)|, where N(x) = {v|xv ? E(G)} is the neighborhood of x. In particular we show: In a 3-connected graph G, if deg S1 + deg S2 ≧ |V(G)| + 1 for each pair of distinct 2-sets of vertices S1, S2 ? V(G), then G is hamiltonian-connected. Several corollaries and related results are also discussed. 相似文献
14.
Fan [G. Fan, Distribution of cycle lengths in graphs, J. Combin. Theory Ser. B 84 (2002) 187-202] proved that if G is a graph with minimum degree δ(G)≥3k for any positive integer k, then G contains k+1 cycles C0,C1,…,Ck such that k+1<|E(C0)|<|E(C1)|<?<|E(Ck)|, |E(Ci)−E(Ci−1)|=2, 1≤i≤k−1, and 1≤|E(Ck)|−|E(Ck−1)|≤2, and furthermore, if δ(G)≥3k+1, then |E(Ck)|−|E(Ck−1)|=2. In this paper, we generalize Fan’s result, and show that if we let G be a graph with minimum degree δ(G)≥3, for any positive integer k (if k≥2, then δ(G)≥4), if dG(u)+dG(v)≥6k−1 for every pair of adjacent vertices u,v∈V(G), then G contains k+1 cycles C0,C1,…,Ck such that k+1<|E(C0)|<|E(C1)|<?<|E(Ck)|, |E(Ci)−E(Ci−1)|=2, 1≤i≤k−1, and 1≤|E(Ck)|−|E(Ck−1)|≤2, and furthermore, if dG(u)+dG(v)≥6k+1, then |E(Ck)|−|E(Ck−1)|=2. 相似文献
15.
Anders Yeo 《Journal of Graph Theory》1999,32(2):123-136
The local irregularity of a digraph D is defined as il(D) = max {|d+ (x) − d− (x)| : x ϵ V(D)}. Let T be a tournament, let Γ = {V1, V2, …, Vc} be a partition of V(T) such that |V1| ≥ |V2| ≥ … ≥ |Vc|, and let D be the multipartite tournament obtained by deleting all the arcs with both end points in the same set in Γ. We prove that, if |V(T)| ≥ max{2il(T) + 2|V1| + 2|V2| − 2, il(T) + 3|V1| − 1}, then D is Hamiltonian. Furthermore, if T is regular (i.e., il(T) = 0), then we state slightly better lower bounds for |V(T)| such that we still can guarantee that D is Hamiltonian. Finally, we show that our results are best possible. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 123–136, 1999 相似文献
16.
Let G be a graph of order n with connectivity κ≥3 and let α be the independence number of G. Set σ4(G)= min{∑4
i
=1
d(x
i
):{x
1,x
2,x
3,x
4} is an independent set of G}. In this paper, we will prove that if σ4(G)≥n+2κ, then there exists a longest cycle C of G such that V(G−C) is an independent set of G. Furthermore, if the minimum degree of G is at least α, then G is hamiltonian.
Received: July 31, 1998?Final version received: October 4, 2000 相似文献
17.
Xuding Zhu 《Journal of Graph Theory》2005,48(3):186-209
For 1 ≤ d ≤ k, let Kk/d be the graph with vertices 0, 1, …, k ? 1, in which i ~j if d ≤ |i ? j| ≤ k ? d. The circular chromatic number χc(G) of a graph G is the minimum of those k/d for which G admits a homomorphism to Kk/d. The circular clique number ωc(G) of G is the maximum of those k/d for which Kk/d admits a homomorphism to G. A graph G is circular perfect if for every induced subgraph H of G, we have χc(H) = ωc(H). In this paper, we prove that if G is circular perfect then for every vertex x of G, NG[x] is a perfect graph. Conversely, we prove that if for every vertex x of G, NG[x] is a perfect graph and G ? N[x] is a bipartite graph with no induced P5 (the path with five vertices), then G is a circular perfect graph. In a companion paper, we apply the main result of this paper to prove an analog of Haj?os theorem for circular chromatic number for k/d ≥ 3. Namely, we shall design a few graph operations and prove that for any k/d ≥ 3, starting from the graph Kk/d, one can construct all graphs of circular chromatic number at least k/d by repeatedly applying these graph operations. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 186–209, 2005 相似文献
18.
Anders Yeo 《Journal of Graph Theory》2007,55(4):325-337
A total dominating set, S, in a graph, G, has the property that every vertex in G is adjacent to a vertex in S. The total dominating number, γt(G) of a graph G is the size of a minimum total dominating set in G. Let G be a graph with no component of size one or two and with Δ(G) ≥ 3. In 6 , it was shown that |E(G)| ≤ Δ(G) (|V(G)|–γt(G)) and conjectured that |E(G)| ≤ (Δ(G)+3) (|V(G)|–γt(G))/2 holds. In this article, we prove that holds and that the above conjecture is false as there for every Δ exist Δ‐regular bipartite graphs G with |E(G)| ≥ (Δ+0.1 ln(Δ)) (|V(G)|–γt(G))/2. The last result also disproves a conjecture on domination numbers from 8 . © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 325–337, 2007 相似文献
19.
It is well known that a graph G of order p ≥ 3 is Hamilton-connected if d(u) + d(v) ≥ p + 1 for each pair of nonadjacent vertices u and v. In this paper we consider connected graphs G of order at least 3 for which d(u) + d(v) ≥ |N(u) ∪ N(v) ∪ N(w)| + 1 for any path uwv with uv ∉ E(G), where N(x) denote the neighborhood of a vertex x. We prove that a graph G satisfying this condition has the following properties: (a) For each pair of nonadjacent vertices x, y of G and for each integer k, d(x, y) ≤ k ≤ |V(G)| − 1, there is an x − y path of length k. (b) For each edge xy of G and for each integer k (excepting maybe one k η {3,4}) there is a cycle of length k containing xy. Consequently G is panconnected (and also edge pancyclic) if and only if each edge of G belongs to a triangle and a quadrangle. Our results imply some results of Williamson, Faudree, and Schelp. © 1996 John Wiley & Sons, Inc. 相似文献
20.
For a finite group G, let n(G) denote the number of conjugacy classes of non-subnormal subgroups of G. In this paper, we show that a finite group G satisfying n(G)≤2|π(G)| is solvable, and for a finite non-solvable group G, n(G) = 2|π(G)|+1 if and only if G?A5. 相似文献