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1.
Let G be a simple undirected graph of order n. For an independent set S ? V(G) of k vertices, we define the k neighborhood intersections Si = {v ? V(G)\S|N(v) ∩ S| = i}, 1 ≦ i ≦ k, with si = |Si|. Using the concept of insertible vertices and the concept of neighborhood intersections, we prove the following theorem. 相似文献
2.
For a graph G, we define σ2(G) := min{d(u) + d(v)|u, v ≠ ∈ E(G), u ≠ v}. Let k ≥ 1 be an integer and G be a graph of order n ≥ 3k. We prove if σ2(G) ≥ n + k − 1, then for any set of k independent vertices v
1,...,v
k
, G has k vertex-disjoint cycles C
1,..., C
k
of length at most four such that v
i
∈ V(C
i
) for all 1 ≤ i ≤ k. And show if σ2(G) ≥ n + k − 1, then for any set of k independent vertices v
1,...,v
k
, G has k vertex-disjoint cycles C
1,..., C
k
such that v
i
∈ V(C
i
) for all 1 ≤ i ≤ k, V(C
1) ∪...∪ V(C
k
) = V(G), and |C
i
| ≤ 4 for all 1 ≤ i ≤ k − 1.
The condition of degree sum σ2(G) ≥ n + k − 1 is sharp.
Received: December 20, 2006. Final version received: December 12, 2007. 相似文献
3.
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
4.
Elizabeth C.M. Maritz 《Quaestiones Mathematicae》2018,41(1):49-63
Let Π = {S1, S2, . . . , Sk} be an ordered partition of the vertex set V (G) of a graph G. The partition representation of a vertex v ∈ V (G) with respect to Π is the k-tuple r(v|Π) = (d(v, S1), d(v, S2), . . . , d(v, Sk)), where d(v, S) is the distance between v and a set S. If for every pair of distinct vertices u, v ∈ V (G), we have r(u|Π) ≠ r(v|Π), then Π is a resolving partition and the minimum cardinality of a resolving partition of V (G) is called the partition dimension of G. We study the partition dimension of circulant graphs, which are Cayley graphs of cyclic groups. Grigorious et al. [On the partition dimension of circulant graphs] proved that pd(Cn(1, 2, . . . , t)) ≥ t + 1 for n ≥ 3. We disprove this statement by showing that if t ≥ 4 is even, then there exists an infinite set of values of n, such that . We also present exact values of the partition dimension of circulant graphs with 3 generators. 相似文献
5.
Zverovich I E 《高校应用数学学报(英文版)》2004,19(3):239-244
§1 IntroductionLet G be a graph with vertex-set V(G) ={ v1 ,v2 ,...,vn} .A labeling of G is a bijectionL:V(G)→{ 1,2 ,...,n} ,where L (vi) is the label of a vertex vi.A labeled graph is anordered pair (G,L) consisting of a graph G and its labeling L.Definition1.An increasing nonconsecutive path in a labeled graph(G,L) is a path(u1 ,u2 ,...,uk) in G such thatL(ui) + 1相似文献
6.
A k-ranking of a graph G = (V, E) is a mapping ϕ: V → {1, 2, ..., k} such that each path with end vertices of the same colour c contains an internal vertex with colour greater than c. The ranking number of a graph G is the smallest positive integer k admitting a k-ranking of G. In the on-line version of the problem, the vertices v
1, v
2, ..., v
n
of G arrive one by one in an arbitrary order, and only the edges of the induced graph G[{v
1, v
2, ..., v
i
}] are known when the colour for the vertex v
i
has to be chosen. The on-line ranking number of a graph G is the smallest positive integer k such that there exists an algorithm that produces a k-ranking of G for an arbitrary input sequence of its vertices.
We show that there are graphs with arbitrarily large difference and arbitrarily large ratio between the ranking number and
the on-line ranking number. We also determine the on-line ranking number of complete n-partite graphs. The question of additivity and heredity is discussed as well. 相似文献
7.
PARTITION A GRAPH WITH SMALL DIAMETER INTO TWO INDUCED MATCHINGS 总被引:5,自引:0,他引:5
Yang Aifeng Yuan Jinjiang Dept. of Appl. Math. South China Univ. of Tech. Guangdong China. Dept. of Math. Zhengzhou Univ. Henan China. 《高校应用数学学报(英文版)》2004,19(3):245-251
§1 IntroductionGraphs considered in this paper are finite and simple.For a graph G,its vertex setandedge set are denoted by V(G) and E(G) ,respectively.If vertices u and v are connected inG,the distance between u and v,denoted by d G(u,v) ,is the length ofa shortest(u,v) -pathin G.The diameter of a connected graph G is the maximum distance between two verticesof G.For X V(G) ,the neighbor set NG(X) of X is defined byNG(X) ={ y∈V(G) \X:there is x∈X such thatxy∈E(G) } .NG({ x} )… 相似文献
8.
Yoshimi Egawa Hikoe Enomoto Ralph J. Faudree Hao Li Ingo Schiermeyer 《Journal of Graph Theory》2003,43(3):188-198
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 vi ∈ V(Ci) for (1 ≤ i ≤ k) if or 3k + 1 ≤ n ≤ 4k, or 4k ≤ n ≤ 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
9.
Jian-Hua Yin 《Czechoslovak Mathematical Journal》2009,59(2):481-487
Let r ≥ 3, n ≥ r and π = (d
1, d
2, ..., d
n
) be a non-increasing sequence of nonnegative integers. If π has a realization G with vertex set V (G) = {v
1, v
2, ..., v
n
} such that d
G
(v
i
) = d
i
for i = 1, 2, ..., n and v
1
v
2 ... v
r
v
1 is a cycle of length r in G, then π is said to be potentially C
r
″-graphic. In this paper, we give a characterization for π to be potentially C
r
″-graphic.
This work was supported by the grant of National Natural Science Foundation of China No. 10861006 and China Scholarship Council. 相似文献
10.
Jiuying Dong 《Journal of Applied Mathematics and Computing》2010,34(1-2):485-493
The theory of vertex-disjoint cycles and 2-factor of graphs has important applications in computer science and network communication. For a graph G, let σ 2(G):=min?{d(u)+d(v)|uv ? E(G),u≠v}. In the paper, the main results of this paper are as follows:
- Let k≥2 be an integer and G be a graph of order n≥3k, if σ 2(G)≥n+2k?2, then for any set of k distinct vertices v 1,…,v k , G has k vertex-disjoint cycles C 1,C 2,…,C k of length at most four such that v i ∈V(C i ) for all 1≤i≤k.
- Let k≥1 be an integer and G be a graph of order n≥3k, if σ 2(G)≥n+2k?2, then for any set of k distinct vertices v 1,…,v k , G has k vertex-disjoint cycles C 1,C 2,…,C k such that:
- v i ∈V(C i ) for all 1≤i≤k.
- V(C 1)∪???∪V(C k )=V(G), and
- |C i |≤4, 1≤i≤k?1.
11.
12.
A digraph G = (V, E) is primitive if, for some positive integer k, there is a u → v walk of length k for every pair u, v of vertices of V. The minimum such k is called the exponent of G, denoted exp(G). The exponent of a vertex u ∈ V, denoted exp(u), is the least integer k such that there is a u → v walk of length k for each v ∈ V. For a set X ⊆ V, exp(X) is the least integer k such that for each v ∈ V there is a X → v walk of length k, i.e., a u → v walk of length k for some u ∈ X. Let F(G, k) : = max{exp(X) : |X| = k} and F(n, k) : = max{F(G, k) : |V| = n}, where |X| and |V| denote the number of vertices in X and V, respectively. Recently, B. Liu and Q. Li proved F(n, k) = (n − k)(n − 1) + 1 for all 1 ≤ k ≤ n − 1. In this article, for each k, 1 ≤ k ≤ n − 1, we characterize the digraphs G such that F(G, k) = F(n, k), thereby answering a question of R. Brualdi and B. Liu. We also find some new upper bounds on the (ordinary) exponent of G in terms of the maximum outdegree of G, Δ+(G) = max{d+(u) : u ∈ V}, and thus obtain a new refinement of the Wielandt bound (n − 1)2 + 1. © 1998 John Wiley & Sons, Inc. J. Graph Theory 28: 215–225, 1998 相似文献
13.
Intersection theorems with geometric consequences 总被引:3,自引:0,他引:3
In this paper we prove that ifℱ is a family ofk-subsets of ann-set, μ0, μ1, ..., μs are distinct residues modp (p is a prime) such thatk ≡ μ0 (modp) and forF ≠ F′ ≠ℱ we have |F ∩F′| ≡ μi (modp) for somei, 1 ≦i≦s, then |ℱ|≦(
s
n
).
As a consequence we show that ifR
n
is covered bym sets withm<(1+o(1)) (1.2)
n
then there is one set within which all the distances are realised.
It is left open whether the same conclusion holds for compositep. 相似文献
14.
A graph G is a k-sphere graph if there are k-dimensional real vectors v
1,…,v
n
such that ij∈E(G) if and only if the distance between v
i
and v
j
is at most 1. A graph G is a k-dot product graph if there are k-dimensional real vectors v
1,…,v
n
such that ij∈E(G) if and only if the dot product of v
i
and v
j
is at least 1. 相似文献
15.
Let G be a graph on the vertex set V={x
1, ..., x
n}. Let k be a field and let R be the polynomial ring k[x
1, ..., x
n]. The graph ideal
I(G), associated to G, is the ideal of R generated by the set of square-free monomials x
ixj so that x
i, is adjacent to x
j. The graph G is Cohen-Macaulay over k if R/I(G) is a Cohen-Macaulay ring.
Let G be a Cohen-Macaulay bipartite graph. The main result of this paper shows that G{v} is Cohen-Macaulay for some vertex v in G. Then as a consequence it is shown that the Reisner-Stanley simplicial complex of I(G) is shellable. An example of N. Terai is presented showing these results fail for Cohen-Macaulay non bipartite graphs.
Partially supported by COFAA-IPN, CONACyT and SNI, México. 相似文献
16.
Selina Yo-Ping Chang Justie Su-Tzu Juan Cheng-Kuan Lin Jimmy J. M. Tan Lih-Hsing Hsu 《Annals of Combinatorics》2009,13(1):27-52
A graph G is hamiltonian connected if there exists a hamiltonian path joining any two distinct nodes of G. Two hamiltonian paths and of G from u to v are independent if u = u
1 = v
1, v = u
v(G)
= v
v(G)
, and u
i
≠ v
i
for every 1 < i < v(G). A set of hamiltonian paths, {P
1, P
2, . . . , P
k
}, of G from u to v are mutually independent if any two different hamiltonian paths are independent from u to v. A graph is k mutually independent hamiltonian connected if for any two distinct nodes u and v, there are k mutually independent hamiltonian paths from u to v. The mutually independent hamiltonian connectivity of a graph G, IHP(G), is the maximum integer k such that G is k mutually independent hamiltonian connected. Let n and k be any two distinct positive integers with n–k ≥ 2. We use S
n,k
to denote the (n, k)-star graph. In this paper, we prove that IHP(S
n,k
) = n–2 except for S
4,2 such that IHP(S
4,2) = 1.
相似文献
17.
Mirko Lepović 《Journal of Applied Mathematics and Computing》2003,11(1-2):109-122
A tree is called starlike if it has exactly one vertex of degree greater than two. In [4] it was proved that two starlike treesG andH are cospectral if and only if they are isomorphic. We prove here that there exist no two non-isomorphic Laplacian cospectral starlike trees. Further, letG be a simple graph of ordern with vertex setV(G)={1,2, …,n} and letH={H 1,H 2, ...H n } be a family of rooted graphs. According to [2], the rooted productG(H) is the graph obtained by identifying the root ofH i with thei-th vertex ofG. In particular, ifH is the family of the paths $P_{k_1 } , P_{k_2 } , ..., P_{k_n } $ with the rooted vertices of degree one, in this paper the corresponding graphG(H) is called the sunlike graph and is denoted byG(k 1,k 2, …,k n ). For any (x 1,x 2, …,x n ) ∈I * n , whereI *={0,1}, letG(x 1,x 2, …,x n ) be the subgraph ofG which is obtained by deleting the verticesi 1, i2, …,i j ∈ V(G) (0≤j≤n), provided that $x_{i_1 } = x_{i_2 } = ... = x_{i_j } = 0$ . LetG(x 1,x 2,…, x n] be the characteristic polynomial ofG(x 1,x 2,…, x n ), understanding thatG[0, 0, …, 0] ≡ 1. We prove that $$G[k_1 , k_2 ,..., k_n ] = \Sigma _{x \in ^{I_ * ^n } } \left[ {\Pi _{i = 1}^n P_{k_i + x_i - 2} (\lambda )} \right]( - 1)^{n - (\mathop \Sigma \limits_{i = 1}^n x_i )} G[x_1 , x_2 , ..., x_n ]$$ where x=(x 1,x 2,…,x n );G[k 1,k 2,…,k n ] andP n (γ) denote the characteristic polynomial ofG(k 1,k 2,…,k n ) andP n , respectively. Besides, ifG is a graph with λ1(G)≥1 we show that λ1(G)≤λ1(G(k 1,k 2, ...,k n )) < for all positive integersk 1,k 2,…,k n , where λ1 denotes the largest eigenvalue. 相似文献
18.
For a fixed multigraph H, possibly containing loops, with V(H)={h1, . . . , hk}, we say a graph G is H-linked if for every choice of k vertices v1, . . . , vk in G, there exists a subdivision of H in G such that vi represents hi (for all i). This notion clearly generalizes the concept of k-linked graphs (as well as other properties). In this paper we determine, for a connected multigraph H and for any sufficiently large graph G, a sharp lower bound on δ(G) (depending upon H) such that G is H-linked. 相似文献
19.
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
20.
MingChu Li 《Graphs and Combinatorics》2001,17(4):687-706
Some known results on claw-free graphs are generalized to the larger class of almost claw-free graphs. In this paper, we
prove the following two results and conjecture that every 5-connected almost claw-free graph is hamiltonian. (1). Every 2-connected
almost claw-free graph G∉J on n≤ 4 δ vertices is hamiltonian, where J is the set of all graphs defined as follows: any graph G in J can be decomposed into three disjoint connected subgraphs G
1, G
2 and G
3 such that E
G
(G
i
, G
j
) = {u
i
, u
j
, v
i
v
j
} for i≠j and i,j = 1, 2, 3 (where u
i
≠v
i
∈V(G
i
) for i = 1, 2, 3). Moreover the bound 4δ is best possible, thereby fully generalizing several previous results. (2). Every 3-connected
almost claw-free graph on at most 5δ−5 vertices is hamiltonian, hereby fully generalizing the corresponding result on claw-free
graphs.
Received: September 21, 1998 Final version received: August 18, 1999 相似文献