It is shown that let G is a connected,N 2-locally connected K 1,4 -restricted graph with δ≥6,which does not contain an induced subgraph H isomorphic to one of G 1,G 2 and G 3,then G is hamiltonian.

This paper proves that: if “ G ” is a 3 Connected { K 1.3 }-free graph and each induced subgraph A of “ G ” satisfies (a 1,a 2) , then G is a penconnected graph (Except for some u and v with d(u,v)=1 , there may not be any (u,v)-k path for k =2,3,4).

(3) for every induced subgraph T of G isomorphic to K_(1,3) or K_(1,3) + e,rnin{max{d_H~w(x),d_G~w(y)} : d(x,y) = 2,x,y ∈ V(T)} ≥ c/2.Then G contains either a Hamilton cycle or a cycle of weight at least c.

The tree T=(V_t, E_t)is a bichromatic tree subgraph in a maximal plannar graph G, in which the subgraph induced by V_t is a tree and T is a component of some bichromatic subgraph Gij of some 4-coloring of G.

It is shown that let G is a connected,N 2-locally connected K 1,4 -restricted graph with δ≥6,which does not contain an induced subgraph H isomorphic to one of G 1,G 2 and G 3,then G is hamiltonian.

A graph is said to be K1,n-free,if it contains no K1.n as an induced subgraph. Some sufficient conditions for the existence of [a,b]--factors in K1,n-free simple graphs are given.

For an integer i and an induced subgraph L of graph G, if x,y∈V(L),d L(x,y)=imax{d G(x),d G(y)}|G|/2,then L is called possessing the property D L(i). Let C(G) be the closure of the graph G.

Given n set X_1,…,X_n, a graph G with the vertex set X=U_i~n=_1X_i Called feasible graph for (X_1,…,X_n)if, for each X_i (i=1,…,n), the induced subgrapl G_i=G[X_i] of G with the vertex set X_i is connected.

If G is a graph with induced subgraphs G1and G2, such that G = G1∪G2 and G1∩G2 = K1, we say that G is the pasteof G1 and G2 at v, where v∈V (G1∩G2), denoted by G = G1∨v G2.In this thesis, we focus on the consecutive edge-coloring problem for cacti.

This paper proves that:let G be a 3-onnected K1.3graph,and if every inducde subgraph A, A of G satisfies (a1,a2),then G is panconnected(except for u and v (G)with d(u,v) = l, there may not be(u,v)- path for k=(2,3,4).

=(V,E) is a 2-connected graph, and X is a set of vertices of G such that for every pair x,x' in X, , and the minimum degree of the induced graph >amp;lt;X>amp;gt; is at least 3, then X is covered by one cycle.

In the on-line version of the problem, the vertices v1, v2, ..., vn of G arrive one by one in an arbitrary order, and only the edges of the induced graph G[{v1, v2, ..., vi}] are known when the colour for the vertex vi has to be chosen.

Topology control is the problem of assigning transmission power values to the nodes of an ad hoc network so that the induced graph satisfies some specified property.

For connectivity, prior work on topology control gave a polynomial time algorithm for minimizing the maximum power assigned to any node (such that the induced graph is connected).

A cluster is de ned as a subset of vertices, whose induced graph is connected.

If the maximum degree of any subgraph induced by a neighborhood of G is at most m, then the independence number of G is at least , where fm+1( x) is a function greater than 0$$

To locally complement a simple graphF at one of its verticesv is to replace the subgraph induced byF onn(v)={w:vw is an edge ofF} by the complementary subgraph.

We show that the chromatic index of G is given by , where is the maximum valency of G and is defined as (w(E(S)) being the number of edges in the subgraph induced by S).

It turns out that the matching number of the subgraph induced by the positive edges is the key parameter that allows us to differentiate between polynomially-solvable and hard instances of the problem.

In this paper, we examine a generalized vertex packing problem (GVP-k) in which k ``violations'' to the independent set restriction are permitted, whereby k edges may exist within the subgraph induced by the chosen set of nodes.

On the basis of P. Kelly's theorem, in §1—§5 the writer investigates at large the r. c. from the structural form in which the(n—2)—order derived subgraphs of a n-Points graph G occur in G. With the concepts of the structural matrix, the column-symmetry-preserving rearrangement of a symmetrical matrix, etc., We first establish some proposition equivalent to the r. e. Then, from the froms of the structural matrices, we pick out some classes of graphs (which include the P. Z. Chinn's result[12]as a particular case),...

On the basis of P. Kelly's theorem, in §1—§5 the writer investigates at large the r. c. from the structural form in which the(n—2)—order derived subgraphs of a n-Points graph G occur in G. With the concepts of the structural matrix, the column-symmetry-preserving rearrangement of a symmetrical matrix, etc., We first establish some proposition equivalent to the r. e. Then, from the froms of the structural matrices, we pick out some classes of graphs (which include the P. Z. Chinn's result[12]as a particular case), of which the reconstructions are unique, and the essential diffieulties in the general case from the viewpoint of the structural matrices are analized. In §6, the reconstructions of partial labeled graphs, the problem of uniqueness of coloring graphs and the relationship between them are discussed. In §7 P. Kelly's theorem is extended to the hypergraphs. The problems and conjectures presented in this paper may stimulate a new approach to the r.c. and the problems related to it, and some of them may be of independent meaning in graph theory.

A graph G is called supercompact if distinct vertices have distinct closed neighborhoods. For a supercompact graph G, an edge e is called removable if G-e is supercompact. The subgraph of G induced by all removable edges is denoted by E0(G ) called the edge nucleus of G. A graph G is called summandable if V(G) can be partitioned into two non-void subsets A and B such that G is the join of G[A] and G[B]. For each integer n>l, we define Ln to be the graph with vertex set V(Ln) = {x1 ,…,xn,b1 ,…, bn, bn+1} and...

A graph G is called supercompact if distinct vertices have distinct closed neighborhoods. For a supercompact graph G, an edge e is called removable if G-e is supercompact. The subgraph of G induced by all removable edges is denoted by E0(G ) called the edge nucleus of G. A graph G is called summandable if V(G) can be partitioned into two non-void subsets A and B such that G is the join of G[A] and G[B]. For each integer n>l, we define Ln to be the graph with vertex set V(Ln) = {x1 ,…,xn,b1 ,…, bn, bn+1} and edge set E(Ln) = {xlbi |1≤i≤n} ∪ {b ibj | i≠j}. The results of this paper are the following.Theorem 1. Let G be a connected suparcompact graph with non-void edge nucleus E0. Then |V(G) | ≤2 |V(E0).| - 1,where the equality holds if and only if G is isomorphic to Ln, for some integer n>l.Theorem 2. A graph G is a supercompact summandable graph and E0 is a forest if and only if either G = {x} + P3 or G = (m{x} ∪ nP2) + (m' {x} ∪ n'P2) where m, n,m',n' are non-negative integers with m+n + m' + n'>2, m + n≠0, m'+n'≠0.