Haruhide Matsuda

A neighborhood condition for graphs to have [ a,b ]-factors

Abstract Let G be a graph of order n, and let a and b be integers such that 1⩽ a b . Then we prove that G has an [ a , b ]-factor if the minimum degree δ ( G )⩾ a , n ⩾2( a + b )( a + b −1)/ b and | N G ( x )∪ N G ( y )|⩾ an /( a + b ) for any two non-adjacent vertices x and y of G. This result is best possible in some sense and it is an extension of the result of Li and Cai (A degree condition for a graph to have [ a , b ]-factors, J. Graph Theory 27 (1998) 1–6).

Fan-type results for the existence of [a,b]-factors

Abstract Let 1 ≤ a b be integers and G a graph of order n sufficiently large for a and b. Then G has an [ a , b ] -factor if the minimum degree is at least a and every pair of vertices distance two apart has cardinality of the neighborhood union at least an / ( a + b ) . This lower bound is sharp. As a consequence, we have a Fan-type condition for a graph to have an [ a , b ] -factor.

On a k -Tree Containing Specified Leaves in a Graph

A k-tree of a graph is a spanning tree with maximum degree at most k. We give sufficient conditions for a graph G to have a k-tree with specified leaves: Let k,s, and n be integers such that k≥2, 0≤s≤k, and n≥s+1. Suppose that (1) G is (s+1)-connected and the degree sum of any k independent vertices of G is at least |G|+(k−1)s−1, or (2) G is n-connected and the independence number of G is at most (n−s)(k−1)+1. Then for any s specified vertices of G, G has a k-tree containing them as leaves. We also discuss the sharpness of the results.

Path factors in claw-free graphs

Abstract A graph G is called claw-free if G has no induced subgraph isomorphic to K1,3. We prove that if G is a claw-free graph with minimum degree at least d, then G has a path factor such that the order of each path is at least d+1.

Degree conditions for Hamiltonian graphs to have [a,b]-factors containing a given Hamiltonian cycle

Abstract Let 1⩽ a b be integers and G a Hamiltonian graph of order | G |⩾( a + b )(2 a + b )/ b . Suppose that δ ( G )⩾ a +2 and max { deg G (x), deg G (y)}⩾a|G|/(a+b)+2 for each pair of nonadjacent vertices x and y in G . Then G has an [ a , b ]-factor which is edge-disjoint from a given Hamiltonian cycle. The lower bound on the degree condition is sharp. For the case of odd a = b , there exists a graph satisfying the conditions of the theorem but having no desired factor. As consequences, we have the degree conditions for Hamiltonian graphs to have [ a , b ]-factors containing a given Hamiltonian cycle.

On 2-edge-connected [a,b] -factors of graphs with Ore-type condition

Let a>=2 and t>=2 be two integers. Suppose that G is a 2-edge-connected graph of order |G|>=2(t+1)((a-2)t+a)+t-1 with minimum degree at least a. Then G has a 2-edge-connected [a,at]-factor if every pair of non-adjacent vertices has degree sum at least 2|G|/(1+t). This lower bound is sharp. As a consequence, we have Ore-type conditions for the existence of a 2-edge-connected [a,b]-factor in graphs.

A neighborhood condition for graphs to have [a, b]-factors II

Abstract. Let a, b, m, and t be integers such that 1≤a<b and 1≤t≤⌉(b−m+1)/a⌉. Suppose that G is a graph of order |G| and H is any subgraph of G with the size |E(H)|=m. Then we prove that G has an [a,b]-factor containing all the edges of H if the minimum degree is at least a, |G|>((a+b)(t(a+b−1)−1)+2m)/b, and |NG(x1)∪⋯ ∪NG(xt)|≥(a|G|+2m)/(a+b) for every independent set {x1,…,xt}⊆V(G). This result is best possible in some sense and it is an extension of the result of H. Matsuda (A neighborhood condition for graphs to have [a,b]-factors, Discrete Mathematics 224 (2000) 289–292).

Regular factors containing a given hamiltonian cycle

Let k ≥ 1 be an integer and let G be a graph having a sufficiently large order n. Suppose that kn is even, the minimum degree of G is at least k + 2, and the degree sum of each pair of nonadjacent vertices in G is at least n + α, where α = 3 for odd k and α = 4 for even k. Then G has a k – factor (i.e. a k – regular spanning subgraph) which is edge-disjoint from a given Hamiltonian cycle. The lower bound on the degree condition is sharp. As a consequence, we have an Ore-type condition for graphs to have a k – factor containing a given Hamiltonian cycle.

Spanning k-ended trees of bipartite graphs

Abstract A tree is called a k -ended tree if it has at most k leaves, where a leaf is a vertex of degree one. We prove the following theorem. Let k ≥ 2 be an integer, and let G be a connected bipartite graph with bipartition ( A , B ) such that | A | ≤ | B | ≤ | A | + k − 1 . If σ 2 ( G ) ≥ ( | G | − k + 2 ) / 2 , then G has a spanning k -ended tree, where σ 2 ( G ) denotes the minimum degree sum of two non-adjacent vertices of G . Moreover, the condition on σ 2 ( G ) is sharp. It was shown by Las Vergnas, and Broersma and Tuinstra, independently that if a graph H satisfies σ 2 ( H ) ≥ | H | − k + 1 then H has a spanning k -ended tree. Thus our theorem shows that the condition becomes much weaker if a graph is bipartite.

Partial Parity (g, f )-Factors and Subgraphs Covering Given Vertex Subsets

Abstract. Let G be a graph and W a subset of V(G). Let g,f:V(G)→Z be two integer-valued functions such that g(x)≤f(x) for all x∈V(G) and g(y)≡f(y) (mod 2) for all y∈W. Then a spanning subgraph F of G is called a partial parity (g,f)-factor with respect to W if g(x)≤degF(x)≤f(x) for all x∈V(G) and degF(y)≡f(y) (mod 2) for all y∈W. We obtain a criterion for a graph G to have a partial parity (g,f)-factor with respect to W. Furthermore, by making use of this criterion, we give some necessary and sufficient conditions for a graph G to have a subgraph which covers W and has a certain given property.