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1.
For any natural number k, a graph G is said to be pancyclic mod k if it contains a cycle of every length modulo k. In this paper, we show that every K1,4-free graph G with minimum degree δ( G) k+3 is pancyclic mod k and every claw-free graph G with δ( G) k+1 is pancyclic mod k, which confirms Thomassen's conjecture (J. Graph Theory 7 (1983) 261–271) for claw-free graphs. 相似文献
2.
Let β( G), Γ( G) and IR( G) be the independence number, the upper domination number and the upper irredundance number, respectively. A graph G is calledΓ-perfect if β( H) = Γ( H), for every induced subgraph H of G. A graph G is called IR- perfect if Γ( H) = IR( H), for every induced subgraph H of G. In this paper, we present a characterization of Γ-perfect graphs in terms of a family of forbidden induced subgraphs, and show that the class of Γ-perfect graphs is a subclass of IR-perfect graphs and that the class of absorbantly perfect graphs is a subclass of Γ-perfect graphs. These results imply a number of known theorems on Γ-perfect graphs and IR-perfect graphs. Moreover, we prove a sufficient condition for a graph to be Γ-perfect and IR-perfect which improves a known analogous result. 相似文献
3.
For any positive integer n and any graph G a set D of vertices of G is a distance- n dominating set, if every vertex vV( G)− D has exactly distance n to at least one vertex in D. The distance- n domination number γ =n( G) is the smallest number of vertices in any distance- n dominating set. If G is a graph of order p and each vertex in G has distance n to at least one vertex in G, then the distance- n domination number has the upper bound p/2 as Ore's upper bound on the classical domination number. In this paper, a characterization is given for graphs having distance- n domination number equal to half their order, when the diameter is greater or equal 2 n−1. With this result we confirm a conjecture of Boland, Haynes, and Lawson. 相似文献
4.
The least domination number γ L of a graph G is the minimum cardinality of a dominating set of G whose domination number is minimum. The least point covering number L of G is the minimum cardinality of a total point cover (point cover including every isolated vertex of G) whose total point covering number is minimum. We prove a conjecture of Sampathkumar saying that
in every connected graph of order n 2. We disprove another one saying that γ L L in every graph but instead of it, we establish the best possible inequality
. Finally, in relation with the minimum cardinality γ t of a dominating set without isolated vertices (total dominating set), we prove that the ratio γ L/γ t can be in general arbitrarily large, but remains bounded by
if we restrict ourselves to the class of trees. 相似文献
5.
This paper starts with a discussion of several old and new conjectures about choosability in graphs. In particular, the list-colouring conjecture, that ch′=χ′ for every multigraph, is shown to imply that if a line graph is ( a : b)-choosable, then it is ( ta : tb)-choosable for every positive integer t. It is proved that ch( H2)=χ( H2) for many “small” graphs H, including inflations of all circuits (connected 2-regular graphs) with length at most 11 except possibly length 9; and that ch″( C)=χ″( C) (the total chromatic number) for various multicircuits C, mainly of even order, where a multicircuit is a multigraph whose underlying simple graph is a circuit. In consequence, it is shown that if any of the corresponding graphs H2 or T( C) is ( a : b)-choosable, then it is ( ta : tb)-choosable for every positive integer t. 相似文献
6.
An L(2,1)- coloring of a graph G is a coloring of G's vertices with integers in {0,1,…, k} so that adjacent vertices’ colors differ by at least two and colors of distance-two vertices differ. We refer to an L(2,1)-coloring as a coloring. The span λ( G) of G is the smallest k for which G has a coloring, a span coloring is a coloring whose greatest color is λ( G), and the hole index ρ( G) of G is the minimum number of colors in {0,1,…,λ( G)} not used in a span coloring. We say that G is full-colorable if ρ( G)=0. More generally, a coloring of G is a no-hole coloring if it uses all colors between 0 and its maximum color. Both colorings and no-hole colorings were motivated by channel assignment problems. We define the no-hole span μ( G) of G as ∞ if G has no no-hole coloring; otherwise μ( G) is the minimum k for which G has a no-hole coloring using colors in {0,1,…, k}. Let n denote the number of vertices of G, and let Δ be the maximum degree of vertices of G. Prior work shows that all non-star trees with Δ3 are full-colorable, all graphs G with n=λ(G)+1 are full-colorable, μ(G)λ(G)+ρ(G) if G is not full-colorable and nλ(G)+2, and G has a no-hole coloring if and only if nλ(G)+1. We prove two extremal results for colorings. First, for every m1 there is a G with ρ(G)=m and μ(G)=λ(G)+m. Second, for every m2 there is a connected G with λ(G)=2m, n=λ(G)+2 and ρ(G)=m. 相似文献
7.
An acyclic graphoidal cover of a graph G is a collection ψ of paths in G such that every path in ψ has at least two vertices, every vertex of G is an internal vertex of at most one path in ψ and every edge of G is in exactly one path in ψ. The minimum cardinality of an acyclic graphoidal cover of G is called the acyclic graphoidal covering number of G and is denoted by η a. A path partition of a graph G is a collection P of paths in G such that every edge of G is in exactly one path in P. The minimum cardinality of a path partition of G is called the path partition number of G and is denoted by π. In this paper we determine η a and π for several classes of graphs and obtain a characterization of all graphs with Δ 4 and η a = Δ − 1. We also obtain a characterization of all graphs for which η a = π. 相似文献
8.
For any graph G a set D of vertices of G is a dominating set, if every vertex vV( G)− D has at least one neighbor in D. The domination number γ( G) is the smallest number of vertices in any dominating set. In this paper, a characterization is given for block graphs having a unique minimum dominating set. With this result, we generalize a theorem of Gunther, Hartnell, Markus and Rall for trees. 相似文献
9.
Let 1 a< b be integers and G a Hamiltonian graph of order | G|( a+ b)(2 a+ b)/ b. Suppose that δ( G) a+2 and max{deg G( x), deg G( y)} a| G|/( a+ b)+2 for each pair of nonadjacent vertices x and y in G. Then G has an [ a, b]-factor which is edge-disjoint from a given Hamiltonian cycle. The lower bound on the degree condition is sharp. For the case of odd a= b, there exists a graph satisfying the conditions of the theorem but having no desired factor. As consequences, we have the degree conditions for Hamiltonian graphs to have [ a, b]-factors containing a given Hamiltonian cycle. 相似文献
10.
An acyclic graphoidal cover of a graph G is a collection ψ of paths in G such that every path in ψ has at least two vertices, every vertex of G is an internal vertex of at most one path in ψ and every edge of G is in exactly one path in ψ. The minimum cardinality of an acyclic graphoidal cover of G is called the acyclic graphoidal covering number of G and is denoted by η a. In this paper we characterize the class of graphs G for which η a=Δ−1 where Δ is the maximum degree of a vertex in G. 相似文献
11.
A weighted graph ( G, w) is a graph G together with a positive weight-function on its vertex set w : V( G)→R >0. The weighted domination number γ w( G) of ( G, w) is the minimum weight w( D)=∑ vDw( v) of a set DV( G) such that every vertex xV( D)− D has a neighbor in D. If ∑ vV(G)w( v)=| V( G)|, then we speak of a normed weighted graph. Recently, we proved that for normed weighted bipartite graphs ( G, w) of order n such that neither G nor the complement
has isolated vertices. In this paper we will extend these Nordhaus–Gaddum-type results to triangle-free graphs. 相似文献
12.
A dominating set for a graph G = ( V, E) is a subset of vertices V′ V such that for all v ε V − V′ there exists some u ε V′ for which { v, u} ε E. The domination number of G is the size of its smallest dominating set(s). For a given graph G with minimum size dominating set D, let m1 ( G, D) denote the number of edges that have neither endpoint in D, and let m2 ( G, D) denote the number of edges that have at least one endpoint in D. We characterize the possible values that the pair ( m1 ( G, D), m2 ( G, D)) can attain for connected graphs having a given domination number. 相似文献
13.
We study the concept of strong equality of domination parameters. Let P1 and P2 be properties of vertex subsets of a graph, and assume that every subset of V( G) with property P2 also has property P1. Let ψ 1( G) and ψ 2( G), respectively, denote the minimum cardinalities of sets with properties P1 and P2, respectively. Then ψ 1( G)ψ 2( G). If ψ 1( G)=ψ 2( G) and every ψ 1( G)-set is also a ψ 2( G)-set, then we say ψ 1( G) strongly equals ψ 2( G), written ψ 1( G)≡ψ 2( G). We provide a constructive characterization of the trees T such that γ( T)≡ i( T), where γ( T) and i( T) are the domination and independent domination numbers, respectively. A constructive characterization of the trees T for which γ( T)=γ t( T), where γ t( T) denotes the total domination number of T, is also presented. 相似文献
14.
For a positive integer k, a k-subdominating function of a graph G=( V, E) is a function f : V→{−1,1} such that ∑ uNG[v]f( u)1 for at least k vertices v of G. The k-subdomination number of G, denoted by γ ks( G), is the minimum of ∑ vVf( v) taken over all k-subdominating functions f of G. In this article, we prove a conjecture for k-subdomination on trees proposed by Cockayne and Mynhardt. We also give a lower bound for γ ks( G) in terms of the degree sequence of G. This generalizes some known results on the k-subdomination number γ ks( G), the signed domination number γ s( G) and the majority domination number γ maj( G). 相似文献
15.
Let G be a graph of order n, and let a and b be integers such that 1 a< b. Let δ( G) be the minimum degree of G. Then we prove that if δ( G)( k−1) a, n( a+ b)( k( a+ b)−2)/ b, and | NG( x1) NG( x2) NG( xk)| an/( a+ b) for any independent subset { x1, x2,…, xk} of V( G), where k2, then G has an [ a, b]-factor. This result is best possible in some sense. 相似文献
16.
Let G be a finite undirected graph with no loops or multiple edges. We define the Laplacian matrix of G,Δ( G)by Δ ij= degree of vertex i and Δ ij-1 if there is an edge between vertex i and vertex j. In this paper we relate the structure of the graph G to the eigenvalues of A( G): in particular we prove that all the eigenvalues of Δ( G) are non-negative, less than or equal to the number of vertices, and less than or equal to twice the maximum vertex degree. Precise conditions for equality are given. 相似文献
17.
A graph G = G( V, E) with lists L( v), associated with its vertices v V, is called L-list colourable if there is a proper vertex colouring of G in which the colour assigned to a vertex v is chosen from L( v). We say G is k- choosable if there is at least one L-list colouring for every possible list assignment L with L( v) = k v V( G). Now, let an arbitrary vertex v of G be coloured with an arbitrary colour f of L(v). We investigate whether the colouring of v can be continued to an L-list colouring of the whole graph. G is called free k-choosable if such an L-list colouring exists for every list assignment L (L(v) = k v V(G)), every vertex v and every colour f L(v). We prove the equivalence of the well-known conjecture of Erd
s et al. (1979): “Every planar graph is 5-choosable” with the following conjecture: “Every planar graph is free 5-choosable”. 相似文献
18.
Let G be a simple graph. The size of any largest matching in G is called the matching number of G and is denoted by ν( G). Define the deficiency of G, def( G), by the equation def( G)=| V( G)|−2ν( G). A set of points X in G is called an extreme set if def( G− X)=def( G)+| X|. Let c0( G) denote the number of the odd components of G. A set of points X in G is called a barrier if c0( G− X)=def( G)+| X|. In this paper, we obtain the following: (1) Let G be a simple graph containing an independent set of size i, where i2. If X is extreme in G for every independent set X of size i in G, then there exists a perfect matching in G. (2) Let G be a connected simple graph containing an independent set of size i, where i2. Then X is extreme in G for every independent set X of size i in G if and only if G=(U,W) is a bipartite graph with |U|=|W|i, and |Γ(Y)||U|−i+m+1 for any Y U, |Y|=m (1mi−1). (3) Let G be a connected simple graph containing an independent set of size i, where i2. Then X is a barrier in G for every independent set X of size i in G if and only if G=(U,W) is a bipartite graph with |U|=|W|=i, and |Γ(Y)|m+1 for any Y U, |Y|=m (1mi−1). 相似文献
19.
Let N denote the set of positive integers and Z denote all integers. The (integral) sum graph of a finite subset S N( Z) is the graph ( S, E) with uv ε E if and only if u + v ε S. A graph G is said to be an (integral) sum graph if it is isomorphic to the (integral) sum graph of some S N( Z). The (integral) sum number of a given graph G is the smallest number of isolated nodes which when added to G result in an (integral) sum graph. We show that the integral sum number of a complete graph with n 4 nodes equals 2n − 3, which proves a conjecture of Harary. And we disprove another conjecture of Harary by showing that there are infinitely many trees which are not caterpillars but are integral sum graphs. 相似文献
20.
A graph is called supereulerian if it has a spanning closed trail. Let G be a 2-edge-connected graph of order n such that each minimal edge cut SE( G) with | S|3 satisfies the property that each component of G− S has order at least ( n−2)/5. We prove that either G is supereulerian or G belongs to one of two classes of exceptional graphs. Our results slightly improve earlier results of Catlin and Li. Furthermore, our main result implies the following strengthening of a theorem of Lai within the class of graphs with minimum degree δ4: If G is a 2-edge-connected graph of order n with δ( G)4 such that for every edge xyE( G) , we have max{ d( x), d( y)}( n−2)/5−1, then either G is supereulerian or G belongs to one of two classes of exceptional graphs. We show that the condition δ( G)4 cannot be relaxed. 相似文献
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