In this paper,we study the enhanced hypercube,an attractive variant of the hypercube and obtained by adding some complementary edges from a hypercube,and focus on cycles embedding on the enhanced hypercube with faulty vertices.Let Fv be the set of faulty vertices in the n-dimensional enhanced hypercube Qn,k(n≥3,1≤k≤n-1).When|Fv|=2,we showed that Qn,k-Fv contains a fault-free cycle of every even length from 4 to 2n-4 where n(n≥3)and k have the same parity;and contains a fault-free cycle of every even length from 4 to 2n-4,simultaneously,contains a cycle of every odd length from n-k+2 to 2n-3 where n(≥3)and k have the diferent parity.Furthermore,when|Fv|=fv≤n-2,we prove that there exists the longest fault-free cycle,which is of even length 2n-2fv whether n(n≥3)and k have the same parity or not;and there exists the longest fault-free cycle,which is of odd length 2n-2fv+1 in Qn,k-Fv where n(≥3)and k have the diferent parity.
For a fixed graph F,a graph G is F-saturated if it has no F as a subgraph,but does contain F after the addition of any new edge.The saturation number,sat(n,F),is the minimum number of edges of a graph in the set of all F-saturated graphs with order n.In this paper,we determine the saturation number sat(n,2P3∪tP2)and characterize the extremal graphs for n≥6t+8.
A graph G has the hourglass property if every induced hourglass S(a tree with a degree sequence 22224) contains two non-adjacent vertices which have a common neighbor in G-V(S).For an integer k≥4,a graph G has the single k-cycle property if every edge of G,which does not lie in a triangle,lies in a cycle C of order at most k such that C has at least「|V(C) /2」 edges which do not lie in a triangle,and they are not adjacent.In this paper,we show that every hourglass-free claw-free graph G of δ(G) ≥3 with the single 7-cycle property is Hamiltonian and is best possible;we also show that every claw-free graph G of δ(G) ≥3 with the hourglass property and with single 6-cycle property is Hamiltonian.
We consider even factors with a bounded number of components in the n-times iterated line graphs L^n(G). We present a characterization of a simple graph G such that L^n(G) has an even factor with at most k components, based on the existence of a certain type of subgraphs in G. Moreover, we use this result to give some upper bounds for the minimum number of components of even factors in L^n(G) and also show that the minimum number of components of even factors in L^n(G) is stable under the closure operation on a claw-free graph G, which extends some known results. Our results show that it seems to be NP-hard to determine the minimum number of components of even factors of iterated line graphs. We also propose some problems for further research.
Let Qn,k(n≥3,1≤k≤n-1) be an n-dimensional enhanced hypercube which is an attractive variant of the hypercube and can be obtained by adding some complementary edges,fv and fe be the numbers of faulty vertices and faulty edges,respectively.In this paper,we give three main results.First,a fault-free path P [u,v] of length at least 2n-2fv-1(respectively,2n-2fv-2) can be embedded on Qn,k with fv+fe≤n-1 when d Qn,k(u,v) is odd(respectively,d Qn,k(u,v) is even).Secondly,an Qn,k is(n-2) edgefault-free hyper Hamiltonian-laceable when n(≥3) and k have the same parity.Lastly,a fault-free cycle of length at least 2n-2fv can be embedded on Qn,k with fe≤n-1 and fv+fe≤2n-4.
A graph has exactly two main eigenvalues if and only if it is a 2-walk linear graph. In this paper, we show some structural properties that a 2-walk (a, b)-linear graph holds. According to these properties, we can estimate and characterize more 2-walk linear graphs that have exactly two main eigenvalues.
