On a Conjecture of Micha Perles
On a Conjecture of Micha Perles
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If A is any induced subgraph of P then A will denote the subgraph of G(P) induced by the corresponding vertices of P.
Let T be a d- regular, induced, connected and non-separating subgraph of Pr(P). We show that T must determine a facet of Pr(P).
If r is a subgraph of G(P) (resp. G(P)). Since both T and G(P) (resp. G(P)) are c/-regular graphs. T = G(P) (resp. T = G(P)). Hence T determines a facet of Pr(P). So we may assume that V(T) fl V(P) ^ and V(T) D V(P) / 0.
Let Tp and Tp be the restriction...s of T to P and P respectively. Consider any vertex v of Tp. V is adjacent to only one vertex in P. Also, T is d- regular. Hence v has at least d — 1 neighbors in Tp. We consider two cases.
Case 1 : Each vertex in Tp has exactly d — 1 neighbors in Tp.
Observe that by symmetry each vertex of T P is also (d — l)-valent in T P and that the two subgraphs Tp and T P are copies of each other. We also observe that: 1. Tp is (d — l)-regular.
2. If Tp has more than one connected component, pick one and call it C .
What to read after On a Conjecture of Micha Perles?
You can find similar books in the "Read Also" column, or choose other free books by Nagabhushana Prabhu to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
If A is any induced subgraph of P then A will denote the subgraph of G(P) induced by the corresponding vertices of P.
Let T be a d- regular, induced, connected and non-separating subgraph of Pr(P). We show that T must determine a facet of Pr(P).
If r is a subgraph of G(P) (resp. G(P)). Since both T and G(P) (resp. G(P)) are c/-regular graphs. T = G(P) (resp. T = G(P)). Hence T determines a facet of Pr(P). So we may assume that V(T) fl V(P) ^ and V(T) D V(P) / 0.
Let Tp and Tp be the restriction...s of T to P and P respectively. Consider any vertex v of Tp. V is adjacent to only one vertex in P. Also, T is d- regular. Hence v has at least d — 1 neighbors in Tp. We consider two cases.
Case 1 : Each vertex in Tp has exactly d — 1 neighbors in Tp.
Observe that by symmetry each vertex of T P is also (d — l)-valent in T P and that the two subgraphs Tp and T P are copies of each other. We also observe that: 1. Tp is (d — l)-regular.
2. If Tp has more than one connected component, pick one and call it C .
What to read after On a Conjecture of Micha Perles?
You can find similar books in the "Read Also" column, or choose other free books by Nagabhushana Prabhu to read onlineMoreLess
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