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Spider web networks: a family of optimal, fault tolerant, Hamiltonian bipartite graphs. (English) Zbl 1057.05052
Let $$P(2r)$$, $$r\geq2$$, be a $$2r$$-gonal prisma with two $$2r$$-gonal faces $$[a_1,a_2,\dots,a_{2r}]$$, $$[b_1,b_2,\dots,b_{2r}]$$, and edges $$a_ia_{i+1},b_ib_{i+1}$$ and $$a_ib_i$$, $$i=1,2,\dots,2r$$ (indices modulo $$2r$$). A {spider web network} SW$$(m,n)$$, $$m=2r$$, $$n=2s$$, $$r\geq2$$, $$s\geq1$$, is a graph obtained from $$P(m)=P(2r)$$ by placing $$n$$ vertices (in order) $$c^{(1)}_i,c^{(2)}_i,\dots,c^{(2s)}_i$$ into the edge $$a_ib_i$$ and inserting edges $$c^{(2k-1)}_{2i+1} c^{(2k-1)}_{2i+2}$$ and $$c^{(2k)}_{2i} c^{(2k)}_{2i+1}$$ for every $$i=1,2,\dots,r$$ (indices modulo $$2r$$) and every $$k$$, $$k=1,2,\dots,s$$. Clearly SW$$(m,n)$$ is a bipartite planar Hamiltonian graph. Let $$C$$ and $$D$$ be a bipartition of the vertex set of SW$$(m,n)$$. In this paper it is proved that SW$$(m,n)-e$$ is Hamiltonian for every edge of SW$$(m,n)$$ and SW$$(m,n)-\{c,d\}$$ remains Hamiltonian for any $$c\in C$$ and $$d\in D$$.

##### MSC:
 05C45 Eulerian and Hamiltonian graphs 05C38 Paths and cycles 05C85 Graph algorithms (graph-theoretic aspects) 05C90 Applications of graph theory
##### Keywords:
bipartite; $$1$$-edge Hamiltonian; $$1_p$$-Hamiltonian; optimal
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