Hamiltonian graphs involving distances

Let G be a graph of order n. We show that if G is a 2-connected graph and max{d(u), d(v)} + |N(u) U N(v)| ≥ n for each pair of vertices u, v at distance two, then either G is hamiltonian or G ?3K_n/3 U T₁ U T₂, where n ? O (mod 3), and T₁ and T₂ are the edge sets of two vertex disjoint triangles containing exactly one vertex from each K_n/3. This result generalizes both Fan's and Lindquester's results as well as several others.     

ON A CONJECTURE ON nTH ORDER DEGREE REGULAR GRAPHS

Abstract

For n a positive integer and v a vertex of a graph G, the nth order degree of v in G, denoted by deg_nv, is the number of vertices at distance n from v. The graph G is said to be nth order regular of degree k if, for every vertex v of G, deg_nv = k. The following conjecture due to Alavi, Lick, and Zou is proved: For n ≥ 2, if G is a connected nth order regular graph of degree 1, then G is either a path of length 2n—1 or G has diameter n. Properties of nth order regular graphs of degree k, k ≥ 1, are investigated.     

A lower bound for groupies in graphs

A non-isolated vertex of a graph G is called a groupie if the average degree of the vertices connected to it is larger than or equal to the average degree of the vertices in G. An isolated vertex is a groupie only if all vertices of G are isolated. While it is well known that every graph must contain at least one groupie, the graph K_n − e contains just 2 groupie vertices for n ≥ 2. In this paper we derive a lower bound for the number of groupies which shows, in particular, that any graph with 2 or more vertices must contain at least 2 groupies. © 1996 John Wiley & Sons, Inc.     

Chromaticity of two-trees

The graphs called 2-trees are defined by recursion. The smallest 2-tree is the complete graph on 2 vertices. A 2-tree on n + 1 vertices (where n ≥ 2) is obtained by adding a new vertex adjacent to each of 2 arbitrarily selected adjacent vertices in a 2-tree on n vertices. A graph G is a 2-tree on n(≥2) vertices if and only if its chromatic polynomial is equal to γ(γ - 1)(γ - 2)^n—2.     

一类直径为2的极大平面图的Mostar指数

图G的Mostar指数定义为Mo(G)=∑uv∈Ε(G)|nu-nv|,其中nu表示在G中到顶点u的距离比到顶点v的距离近的顶点个数,nv表示到顶点v的距离比到顶点u的距离近的顶点个数.若一个图G的任两点之间的距离至多为2,且不是完全图,则称G是一个直径为2的图.已知直径为2点数至少为4的极大平面图的最小度为3或4.本文研究了直径为2且最小度为4的极大平面图的Mostar指数.具体说,若G是一个点数为n,直径为2,最小度为4的极大平面图,则(1)当n≤12时,Mostar指数被完全确定;(2)当n≥13时,4/3n2-44/3n+94/3≤Mo(G)≤2n2-16n+24,且达到上,下界的极图同时被找到.     

Doubly stochastic graph matrices,II

If G is a graph on n vertices, its Laplacian matrix L(G) = D(G) - A(G) is the difference of the diagonal matrix of vertex degrees and the adjacency matrix. The main purpose of this note is to continue the study of the positive definite, doubly stochastic graph matrix (I_n + L(G))?¹= ω(G) = (w_ij). If, for example, w(G) = min w_ij, then w(G)≥0 with equality if and only if G is disconnected and w(G) ≤ l/(n + 1) with equality if and only if G = K_n. If i¦j, then w_ii ≥2wij, with equality if and only if the ith vertex has degree n - 1. In a sense made precise in the note, max w,, identifies most remote vertices of G. Relations between these new graph invariants and the algebraic connectivity emerge naturally from the fact that the second largest eigenvalue of ω(G) is 1/(1 + a(G)).     

The ratio of the distance irredundance and domination numbers of a graph

Let n ≥ 1 be an integer and let G be a graph. A set D of vertices in G is defined to be an n-dominating set of G if every vertex of G is within distance n from some vertex of D. The minimum cardinality among all n-dominating sets of G is called the n-domination number of G and is denoted by γ_n(G). A set / of vertices in G is n-irredundant if for every vertex x ∈ / there exists a vertex y that is within distance n from x but at distance greater than n from every vertex of / - {x}. The n-irredundance number of G, denoted by ir_n(G), is the minimum cardinality taken over all maximal n-irredundant sets of vertices of G. We show that inf{ir_n(G)/γ_n(G) | G is an arbitrary finite undirected graph with neither loops nor multiple edges} = 1/2 with the infimum not being attained. Subsequently, we show that 2/3 is a lower bound on all quotients ir_n(T)/γ_n(T) in which T is a tree. Furthermore, we show that, for n ≥ 2, this bound is sharp. These results extend those of R. B. Allan and R.C. Laskar [“On Domination and Some Related Concepts in Graph Theory,” Utilitas Mathematica, Vol. 21 (1978), pp. 43–56], B. Bollobás and E. J. Cockayne [“Graph-Theoretic Parameters Concerning Domination, Independence and Irredundance,” Journal of Graph Theory, Vol. 3 (1979), pp. 241–249], and P. Damaschke [Irredundance Number versus Domination Number, Discrete Mathematics, Vol. 89 (1991), pp. 101–104].     

A generalization of a result of Bauer and Schmeichel

We prove that if G is a 1-tough graph withn ≥ 3 vertices such thatd(u) + d(v) +d(w) ≥n+ κ —2 holds for any triple of independent verticesu, v andw ofG, thenG is hamiltonian, wherek is the vertex connectivity ofG. This generalizes a recent result of Baur and Schmeichel.     

Maximal triangle-free graphs with restrictions on the degrees

We investigate the problem that at least how many edges must a maximal triangle-free graph on n vertices have if the maximal valency is ≤D. Denote this minimum value by F(n, D). For large enough n, we determine the exact value of F(n, D) if D ≥ (n ? 2)/2 and we prove that lim F(n, cn)/n = K(c) exists for all 0 < c with the possible exception of a sequence c_k → 0. The determination of K(c) is a finite problem on all intervals [γ, ∞). For D = cn^?, 1/2 < ? < 1, we give upper and lower bounds for F(n, D) differing only in a constant factor. (Clearly, D < (n - 1)^1/2 is impossible in a maximal triangle-free graph.)     

Isoperimetric numbers of graph bundles

The main aim of this paper is to give some upper and lower bounds for the isoperimetric numbers of graph coverings or graph bundles, with exact values in some special cases. In addition, we show that the isoperimetric number of any covering graph is not greater than that of the base graph. Mohar's theorem for the isoperimetric number of the cartesian product of a graph and a complete graph can be extended to a more general case: The isoperimetric numberi(G × K _2n) of the cartesian product of any graphG and a complete graphK _2n on even vertices is the minimum of the isoperimetric numberi(G) andn, and it is also a sharp lower bound of the isoperimetric numbers of all graph bundles over the graphG with fiberK _2n. Furthermore, ifn 2i(G) then the isoperimetric number of any graph bundle overG with fibreK _n is equal to the isoperimetric numberi(G) ofG. Partially supported by The Ministry of Education, Korea.     

Remarks on Vertex-Distinguishing IE-Total Coloring of Complete Bipartite Graphs K4,n and Kn,n

Let G be a simple graph.An IE-total coloring f of G refers to a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color.Let C(u) be the set of colors of vertex u and edges incident to u under f.For an IE-total coloring f of G using k colors,if C(u)=C(v) for any two different vertices u and v of V(G),then f is called a k-vertex-distinguishing IE-total-coloring of G,or a k-VDIET coloring of G for short.The minimum number of colors required for a VDIET coloring of G is denoted by χ ie vt (G),and it is called the VDIET chromatic number of G.We will give VDIET chromatic numbers for complete bipartite graph K4,n (n≥4),K n,n (5≤ n ≤ 21) in this article.     

A CHARACTERIZATION OF STABLE GRAPHS ON A MAXIMUM(MINIMUM) NUMBER OF EDGES

ABSTRACT

A connected, nontrivial, simple graph of order v is said to be α,β destructible if α,β are factors of v and an α-set of edges, E', exists whose removal from G isolates exactly the vertices in α,β-set V'. Graphs which are not α,β destructible for any α,β are called stable, If G is a stable graph on a prime number p ≥ 7 of vertices, then we show that G has a maximum number of edges if and only if G is K_2,p-2, We also characterize stable graphs on a minimum number of edges.     

TWO RESULTS ON GENERALIZED CHROMATIC NUMBERS

Abstract

The m'th chromatic number X_m(G) of a graph G is the least number of colours required to colour the vertices of G such that no m-clique of G is mono-coloured. For each k ≥ 2 and m ≥ 2 we determine for which r a graph G with m'th chromatic number k and clique number r exists. We also determine for which n a graph G exists with X_n (G + K) = X_m (G) = k.     

Zero-Divisor Semigroups and Some Simple Graphs

In this article, we study commutative zero-divisor semigroups determined by graphs. We prove that for all n ≥ 4, the complete graph K _n together with two end vertices has a unique corresponding zero-divisor semigroup, while the complete graph K _n together with three end vertices has no corresponding semigroups. We determine all the twenty zero-divisor semigroups whose zero-divisor graphs are the complete graph K ₃ together with an end vertex.     

可圈的一个充分条件

本文我们证明如下结果:设G=(V,E)是一个n(n≥3)阶k-连通(k≥2)图,记X1,X2,…,Xk为V的子集,X=X1∪X2∪…∪Xk.若对每个I,I=1,2,…,k,满足:对任意的u,v∈Xi,有d(u) d(v)≥n或|N(u)∪N(v)|≥n-δ或|N(u)∩N(v)|≥α,这里δ是G的最小度,α是G的独立数,则G是X-可圈的.     

Domination number in graphs with minimum degree two

A set D of vertices of a graph G = （V, E） is called a dominating set if every vertex of V not in D is adjacent to a vertex of D. In 1996, Reed proved that every graph of order n with minimum degree at least 3 has a dominating set of cardinality at most 3n/8. In this paper we generalize Reed＇s result. We show that every graph G of order n with minimum degree at least 2 has a dominating set of cardinality at most （3n ＋IV21）/8, where V2 denotes the set of vertices of degree 2 in G. As an application of the above result, we show that for k ≥ 1, the k-restricted domination number rk （G, γ） ≤ （3n＋5k）/8 for all graphs of order n with minimum degree at least 3.     

On graphs with prescribed median I

The distance of a vertex u in a connected graph H is the sum of all the distances from u to the other vertices of H. The median M(H) of H is the subgraph of H induced by the vertices of minimum distance. For any graph G, let f(G) denote the minimum order of a connected graph H satisfying M(H) ? G. It is shown that if G has n vertices and minimum degree δ then f(G) ? 2n ? δ + 1. Graphs having both median and center prescribed are constructed. It is also shown that if the vertices of a K_r are removed from a graph H, then at most r components of the resulting graph contain median vertices of H.     

Complete subgraphs with large degree sums

Let t(n, k) denote the Turán number—the maximum number of edges in a graph on n vertices that does not contain a complete graph K_k+1. It is shown that if G is a graph on n vertices with n ≥ k²(k – 1)/4 and m < t(n, k) edges, then G contains a complete subgraph K_k such that the sum of the degrees of the vertices is at least 2km/n. This result is sharp in an asymptotic sense in that the sum of the degrees of the vertices of K_k is not in general larger, and if the number of edges in G is at most t(n, k) – ? (for an appropriate ?), then the conclusion is not in general true. © 1992 John Wiley & Sons, Inc.     

UPPER BOUNDS ON THE LOWER OPEN PACKING NUMBER OF A TREE

Abstract

Let G be a graph and let v be a vertex of G. The open neigbourhood N(v) of v is the set of all vertices adjacent with v in G. An open packing of G is a set of vertices whose open neighbourhoods are pairwise disjoint. The lower open packing number of G, denoted ρ^° _L(G), is the minimum cardinality of a maximal open packing of G while the (upper) open packing number of G, denoted ρ^°(G), is the maximum cardinality among all open packings of G. It is known (see [7]) that if G is a connected graph of order n ≥3, then ρ^°(G) ≤ 2n/3 and this bound is sharp (even for trees). As a consequence of this result, we know that ρ^° _L(G) ≤ 2n/3. In this paper, we improve this bound when G is a tree. We show that if G is a tree of order n with radius 3, then ρ^° _L(G) ≤ n/2 + 2 √n-1, and this bound is sharp, while if G is a tree of order n with radius at least 4, then ρ^° _L(G) is bounded above by 2n/3—O√n).     

Maximum Zagreb index, minimum hyper-Wiener index and graph connectivity

In this work we show that among all n-vertex graphs with edge or vertex connectivity k, the graph G=K_k(K₁+K_n−k−1), the join of K_k, the complete graph on k vertices, with the disjoint union of K₁ and K_n−k−1, is the unique graph with maximum sum of squares of vertex degrees. This graph is also the unique n-vertex graph with edge or vertex connectivity k whose hyper-Wiener index is minimum.