Characterizations On Potentially A_r+1- H -graphic Sequences

Let G be a graph, with vertex set V (?) {v 1 , v2 , (?) vn } and denote the degree of vertex vi by d i, the sequence (?) (?) ( d₁ , d₂, (?) dn) is called a degree sequence of G . A non-increasing sequence (?) (?) ( d₁ , d₂, (?) dn) is graphic if there is a simple graph G with degree sequence (?) , then G is called a realization of (?) . If there exists a realization of (?) (?) ( d₁ , d₂, (?) dn) contains H as a subgraph, that is H (?)G, then we say that (?) is potentiallyH -graphic. If E ( H ) is edges of H , we may obtain a graph by deleting E ( H ) from G but leaving the vertices and the remaining edges intact. The resulting graph is denoted by K r(?)1 (?)H and the degree sequence of K_r+1-H is denoted by (?) (?) ( d₁(?) , d₂(?) , (?) , dr(?)1(?)). (?) is potentially A_r+1-H-graphic if G is a realization of (?) with its vertex set V ( G ) (?) {v 1 , v2 , (?) , vn} satisfies d G ( vi ) (?) di, 1 (?) i (?) n, G[ {v 1 , v2 , (?) , vr (?)1 }](?) K r(?)1(?)H such that d H ( vi ) (?) di(?), 1 (?) i (?) r(?) 1. The K k is the graph of k vertices in which any two vertices are adjacent. A cycle of length k is called a k -cycle, denoted by C k. For 12n(?) m(?) ,let C m ,ndenote the graph K m (?) ( K mc (?) K n (?)2m). In this thesis,we characterize the potentially C 2,6-graphic sequ- ences. Then we also give characterizations for (?) (?) ( d₁ , d₂, (?) dn)to be potentially Ar (?)1 (?) mK2 and Ar (?)1 (?) tCk- graphic.