Tomoki Yamashita

Note: On degree sum conditions for long cycles and cycles through specified vertices

Let G be a graph. For [emailxa0protected]?V(G), let @Dk(S) denote the maximum value of the degree sums of the subsets of S of order k. In this paper, we prove the following two results. (1) Let G be a 2-connected graph. If @D2(S)>=d for every independent set S of order @k(G)+1, then G has a cycle of length at least min{d,|V(G)|}. (2) Let G be a 2-connected graph and X a subset of V(G). If @D2(S)>=|V(G)| for every independent set S of order @k(X)+1 in G[X], then G has a cycle that includes every vertex of X. This suggests that the degree sum of nonadjacent two vertices is important for guaranteeing the existence of these cycles.

Spanning Trees with Few Leaves

In this paper, we prove that an m-connected graph G on n vertices has a spanning tree with at most k leaves (for k ≥ 2 and m ≥ 1) if every independent set of G with cardinality mxa0+xa0k contains at least one pair of vertices with degree sum at least nxa0−xa0kxa0+xa01. This is a common generalization of results due to Broersma and Tuinstra and to Win.

Degree sum and connectivity conditions for dominating cycles

For a graph G, let @sk(G) be the minimum degree sum of an independent set of k vertices. Ore showed that if G is a graph of order n>=3 with @s2(G)>=n then G is hamiltonian. Let @k(G) be the connectivity of a graph G. Bauer, Broersma, Li and Veldman proved that if G is a 2-connected graph on n vertices with @s3(G)>[emailxa0protected](G), then G is hamiltonian. On the other hand, Bondy showed that if G is a 2-connected graph on n vertices with @s3(G)>=n+2, then each longest cycle of G is a dominating cycle. In this paper, we prove that if G is a 3-connected graph on n vertices with @s4(G)>[emailxa0protected](G)+3, then G contains a longest cycle which is a dominating cycle.

On a Spanning Tree with Specified Leaves

Let k ≥ 2 be an integer. We show that if G is a (kxa0+xa01)-connected graph and each pair of nonadjacent vertices in G has degree sum at least |G|xa0+xa01, then for each subset S of V(G) with |S|xa0=xa0k, G has a spanning tree such that S is the set of endvertices. This result generalizes Ore’s theorem which guarantees the existence of a Hamilton path connecting any two vertices.

Degree conditions on claws and modified claws for hamiltonicity of graphs

Ore presented a degree condition involving every pair of nonadjacent vertices for a graph to be hamiltonian. Fan [New sufficient conditions for cycles in graphs, J. Combin. Theory Ser. B 37 (1984) 221-227] showed that not all the pairs of nonadjacent vertices are required, but only the pairs of vertices at the distance two suffice. Bedrossian et al. [A generalization of Fans condition for hamiltonicity, pancyclicity, and hamiltonian connectedness, Discrete Math. 115 (1993) 39-50] improved Fans result involving the pairs of vertices contained in an induced claw or an induced modified claw. On the other hand, Matthews and Sumner [Longest paths and cycles in K1,3-free graphs, J. Graph Theory 9 (1985) 269-277] gave a minimum degree condition for a claw-free graph to be hamiltonian. In this paper, we give a new degree condition in an induced claw or an induced modified claw ensuring the hamiltonicity of graphs which extends both results of Bederossian et al. and Matthews and Sumner.

Long cycles in graphs without hamiltonian paths

For a graph G, p(G) and c(G) denote the order of a longest path and a longest cycle of G, respectively. Bondy and Locke [J.A. Bondy, S.C. Locke, Relative length of paths and cycles in 3-connected graphs, Discrete Math. 33 (1981) 111-122] consider the gap between p(G) and c(G) in 3-connected graphs G. Starting with this result, there are many results appeared in this context, see [H. Enomoto, J. van den Heuvel, A. Kaneko, A. Saito, Relative length of long paths and cycles in graphs with large degree sums, J. Graph Theory 20 (1995) 213-225; M. Lu, H. Liu, F. Tian, Relative length of longest paths and cycles in graphs, Graphs Combin. 23 (2007) 433-443; K. Ozeki, M. Tsugaki, T. Yamashita, On relative length of longest paths and cycles, preprint; I. Schiermeyer, M. Tewes, Longest paths and longest cycles in graphs with large degree sums, Graphs Combin. 18 (2002) 633-643]. In this paper, we investigate graphs G with p(G)-c(G) at most 1 or at most 2, but with no hamiltonian paths. Let G be a 2-connected graph of order n, which has no hamiltonian paths. We show two results as follows: (i) if @s4(G)>=13(4n-2), then p(G)-c(G)@?1, and (ii) if @s4(G)>=n+3, then p(G)-c(G)@?2.

Cycles within specified distance from each vertex

Abstract Let G be a graph and let f be a non-negative integer-valued function defined on V(G). Then a cycle C is called an f-dominating cycle if dG(v,C)⩽f(v) holds for each v∈V(G), where dG(v,C) denotes the distance between v and C. A set S is called an f-stable set if dG(u,v)⩾f(u)+f(v) holds for each pair of distinct vertices u, v in S, and denote by αf(G) the order of a largest f-stable set in G. In this paper, we prove that if a k-connected graph G (k⩾2) satisfies αf+1(G)⩽k, then G has an f-dominating cycle, where f+1 is the function defined by (f+1)(v)=f(v)+1. By taking an appropriate function as f, we can deduce a number of known results from this theorem.

Hamiltonian cycles and dominating cycles passing through a linear forest

Let G be an (m+2)-graph on n vertices, and F be a linear forest in G with |E(F)|=m and @w1(F)=s, where @w1(F) is the number of components of order one in F. We denote by @s3(G) the minimum value of the degree sum of three vertices which are pairwise non-adjacent. In this paper, we give several @s3 conditions for a dominating cycle or a hamiltonian cycle passing through a linear forest. We first prove that if @s3(G)>=n+2m+2+max{s-3,0}, then every longest cycle passing through F is dominating. Using this result, we prove that if @s3(G)>=n+@k(G)+2m-1 then G contains a hamiltonian cycle passing through F. As a corollary, we obtain a result that if G is a 3-connected graph and @s3(G)>=n+@k(G)+2, then G is hamiltonian-connected.

A degree sum condition for long cycles passing through a linear forest

Let G be a (k+m)-connected graph and F be a linear forest in G such that |E(F)|=m and F has at most k-2 components of order 1, where k>=2 and m>=0. In this paper, we prove that if every independent set S of G with |S|=k+1 contains two vertices whose degree sum is at least d, then G has a cycle C of length at least min{d-m,|V(G)|} which contains all the vertices and edges of F.

Edge-dominating cycles in graphs

A set S of vertices in a graph G is said to be an edge-dominating set if every edge in G is incident with a vertex in S. A cycle in G is said to be a dominating cycle if its vertex set is an edge-dominating set. Nash-Williams [Edge-disjoint hamiltonian circuits in graphs with vertices of large valency, Studies in Pure Mathematics, Academic Press, London, 1971, pp. 157-183] has proved that every longest cycle in a 2-connected graph of order n and minimum degree at least 13(n+2) is a dominating cycle. In this paper, we prove that for a prescribed positive integer k, under the same minimum degree condition, if n is sufficiently large and if we take k disjoint cycles so that they contain as many vertices as possible, then these cycles form an edge-dominating set. Nash-Williams Theorem corresponds to the case of k=1 of this result.