Masao Tsugaki

Improved Upper Bounds for Gallai-Ramsey Numbers of Paths and Cycles

Given a graph G and a positive integer k, define the Gallai-Ramsey number to be the minimum number of vertices n such that any k-edge-coloring of Kn contains either a rainbow (all different colored) triangle or a monochromatic copy of G. In this work, we improve upon known upper bounds on the Gallai-Ramsey numbers for paths and cycles. All these upper bounds now have the best possible order of magnitude as functions of k.

A note on a spanning 3-tree

A tree T is called a k-tree, if the maximum degree of T is at most k. In this paper, we prove that if G is an n-connected graph with independence number at most n + m + 1 (n≥1,n≥m≥0), then G has a spanning 3-tree T with at most m vertices of degree 3.

An Anti-Ramsey Theorem on Diamonds

AbstractLet G be the diamond (the graph obtained from K4 by deleting an edge) and, for every n ≥ 4, let f(n, G) be the minimum integer k such that, for every edge-coloring of the complete graph of order n which uses exactly k colors, there is at least one copy of G all whose edges have different colors. Let ext(n, {C3, C4}) be the maximum number of edges of a graph on n vertices free of triangles and squares. Here we prove that for every n ≥ 4,

Spanning k-ended trees of bipartite graphs

{\rm {ext}}(n, \{C_3, C_4\})+ 2\leq f(n,G)\leq {\rm {ext}}(n, \{C_3,C_4\})+ (n+1).

Pairs of forbidden class of subgraphs concerning K 1, 3 and P 6 to have a cycle containing specified vertices.

In [H. Broersma, H. Li, J. Li, F. Tian, H.J. Veldman, Cycles through subsets with large degree sums, Discrete Math. 171 (1997) 43-54], Duffus et al. showed that every connected graph G which contains no induced subgraph isomorphic to a claw or a net is traceable. They also showed that if a 2-connected graph G satisfies the above conditions, then G is hamiltonian. In this paper, modifying the conditions of Duffus et al.s theorems, we give forbidden structures for a specified set of vertices which assures the existence of paths and cycles passing through these vertices.

Spanning Trees with Few Leaves

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.

On relative length of longest paths and cycles

Las Vergnas (CR Acad Sci Paris Sér A 272:1297–1300, 1971), and Broersma and Tuinstra (J Graph Theory 29:227–237, 1998) independently investigated a degree sum condition for a graph to have a spanning tree whose number of leaves is restricted. In this paper, we obtain the bipartite analogy of this result.