The 2-sum of two digraphs and , denoted , is the digraph obtained from the disjoint union of and by identifying an arc in with an arc in . A digraph D is supereulerian if D contains a spanning eulerian subdigraph. It ...The 2-sum of two digraphs and , denoted , is the digraph obtained from the disjoint union of and by identifying an arc in with an arc in . A digraph D is supereulerian if D contains a spanning eulerian subdigraph. It has been noted that the 2-sum of two supereulerian (or even hamiltonian) digraphs may not be supereulerian. We obtain several sufficient conditions on and for to be supereulerian. In particular, we show that if and are symmetrically connected or partially symmetric, then is supereulerian.展开更多
Letk?2 be an integer and let G be a graph of ordern with minimum degree at leastk, n?8k -16 for evenn and n?6k - 13 for oddn. If the degree sum of each pair of nonadjacent vertices of G is at least n, then for any giv...Letk?2 be an integer and let G be a graph of ordern with minimum degree at leastk, n?8k -16 for evenn and n?6k - 13 for oddn. If the degree sum of each pair of nonadjacent vertices of G is at least n, then for any given Hamiltonian cycleC. G has a [k, k + 1]-factor containingC展开更多
对于一个整数.s≥0,如果图G的任何一个点子集S (?) V(G)满足|S|≤s,并且G-S是哈密尔顿的,那么称图G是s-哈密尔顿的.本文证明原图是平面图的4-连通线图是2-哈密尔顿的并且还是哈密尔顿连通的.这一结果推广了赖虹建在[Graph and Combinato...对于一个整数.s≥0,如果图G的任何一个点子集S (?) V(G)满足|S|≤s,并且G-S是哈密尔顿的,那么称图G是s-哈密尔顿的.本文证明原图是平面图的4-连通线图是2-哈密尔顿的并且还是哈密尔顿连通的.这一结果推广了赖虹建在[Graph and Combinatorics,1994, 10:249-253]中的结果.展开更多
Given a digraph D =（V, A）, the competition graph G of D, denoted by C（D）, has the same set of vertices as D and an edge between vertices x and y if and only if N;（x）∩N;（y）≠Ф. In this paper, we investigate t...Given a digraph D =（V, A）, the competition graph G of D, denoted by C（D）, has the same set of vertices as D and an edge between vertices x and y if and only if N;（x）∩N;（y）≠Ф. In this paper, we investigate the competition graphs of round digraphs and give a necessary and sufficient condition for these graphs to be hamiltonian.展开更多
Let σk（G） denote the minimum degree sum of k independent vertices in G and α（G） denote the number of the vertices of a maximum independent set of G. In this paper we prove that if G is a 4-connected graph of ord...Let σk（G） denote the minimum degree sum of k independent vertices in G and α（G） denote the number of the vertices of a maximum independent set of G. In this paper we prove that if G is a 4-connected graph of order n and σ5（G） 〉 n ＋ 3σ（G） ＋ 11, then G is Hamiltonian.展开更多
Let G be a multigraph.Suppose that e=u1v1 and e′=u2v2 are two edges of G.If e≠e′,then G(e,e′)is the graph obtained from G by replacing e=u1v1 with a path u1vev1 and by replacing e′=u2v2 with a path u2ve′v2,where...Let G be a multigraph.Suppose that e=u1v1 and e′=u2v2 are two edges of G.If e≠e′,then G(e,e′)is the graph obtained from G by replacing e=u1v1 with a path u1vev1 and by replacing e′=u2v2 with a path u2ve′v2,where ve,ve′are two new vertices not in V(G).If e=e′,then G(e,e′),also denoted by G(e),is obtained from G by replacing e=u1v1 with a path u1vev1.A graph G is strongly spanning trailable if for any e,e′∈E(G),G(e,e′)has a spanning(ve,ve′)-trail.The design of n processor network with given number of connections from each processor and with a desirable strength of the network can be modelled as a degree sequence realization problem with certain desirable graphical properties.A sequence d=(d1,d2,⋯,dn)is multigraphic if there is a multigraph G with degree sequence d,and such a graph G is called a realization of d.A multigraphic degree sequence d is strongly spanning trailable if d has a realization G which is a strongly spanning trailable graph,and d is line-hamiltonian-connected if d has a realization G such that the line graph of G is hamiltonian-connected.In this paper,we prove that a nonincreasing multigraphic sequence d=(d1,d2)⋯,dn)is strongly spanning trailable if and only if either n=1 and d1=0 or n≥2 and dn≥3.Applying this result,we prove that for a nonincreasing multigraphic sequence d=(d1,d2,⋯,dn),if n≥2 and dn≥3,then d is line-hamiltonian-connected.展开更多
文摘The 2-sum of two digraphs and , denoted , is the digraph obtained from the disjoint union of and by identifying an arc in with an arc in . A digraph D is supereulerian if D contains a spanning eulerian subdigraph. It has been noted that the 2-sum of two supereulerian (or even hamiltonian) digraphs may not be supereulerian. We obtain several sufficient conditions on and for to be supereulerian. In particular, we show that if and are symmetrically connected or partially symmetric, then is supereulerian.
文摘Letk?2 be an integer and let G be a graph of ordern with minimum degree at leastk, n?8k -16 for evenn and n?6k - 13 for oddn. If the degree sum of each pair of nonadjacent vertices of G is at least n, then for any given Hamiltonian cycleC. G has a [k, k + 1]-factor containingC
基金Supported by NSFC(11401353)TYAL of ShanxiNatural Science Foundation of Shanxi Province(2016011005)
文摘Given a digraph D =（V, A）, the competition graph G of D, denoted by C（D）, has the same set of vertices as D and an edge between vertices x and y if and only if N;（x）∩N;（y）≠Ф. In this paper, we investigate the competition graphs of round digraphs and give a necessary and sufficient condition for these graphs to be hamiltonian.
基金Supported by NNSF of China (Grant No. 60373012)supported by NSFC (Grant No. 10601044)XJEDU2006S05
文摘Let σk（G） denote the minimum degree sum of k independent vertices in G and α（G） denote the number of the vertices of a maximum independent set of G. In this paper we prove that if G is a 4-connected graph of order n and σ5（G） 〉 n ＋ 3σ（G） ＋ 11, then G is Hamiltonian.
基金This paper is supported by the National Natural Science Foundation of China(Nos.11771039,11971054)Fundamental Research Funds for the Central Universities of China(No.2015JBM107)the 111 Project of China(No.B16002)。
文摘Let G be a multigraph.Suppose that e=u1v1 and e′=u2v2 are two edges of G.If e≠e′,then G(e,e′)is the graph obtained from G by replacing e=u1v1 with a path u1vev1 and by replacing e′=u2v2 with a path u2ve′v2,where ve,ve′are two new vertices not in V(G).If e=e′,then G(e,e′),also denoted by G(e),is obtained from G by replacing e=u1v1 with a path u1vev1.A graph G is strongly spanning trailable if for any e,e′∈E(G),G(e,e′)has a spanning(ve,ve′)-trail.The design of n processor network with given number of connections from each processor and with a desirable strength of the network can be modelled as a degree sequence realization problem with certain desirable graphical properties.A sequence d=(d1,d2,⋯,dn)is multigraphic if there is a multigraph G with degree sequence d,and such a graph G is called a realization of d.A multigraphic degree sequence d is strongly spanning trailable if d has a realization G which is a strongly spanning trailable graph,and d is line-hamiltonian-connected if d has a realization G such that the line graph of G is hamiltonian-connected.In this paper,we prove that a nonincreasing multigraphic sequence d=(d1,d2)⋯,dn)is strongly spanning trailable if and only if either n=1 and d1=0 or n≥2 and dn≥3.Applying this result,we prove that for a nonincreasing multigraphic sequence d=(d1,d2,⋯,dn),if n≥2 and dn≥3,then d is line-hamiltonian-connected.
