Hikoe Enomoto

Choice number of some complete multi-partite graphs

One of the authors has conjectured that every graph G with 2χ(G) + 1 or fewer vertices is χ(G)-choosable. Motivated by this, we investigate the choice numbers of some complete k-partite graphs of order slightly larger than 2k, and settle the conjecture for some special cases. We also present several complete multi-partite graphs whose choice numbers are not equal to their chromatic numbers.

Partition of a graph into cycles and degenerated cycles

Abstract Let G be a graph of order n and k any positive integer with k ⩽ n . It has been shown by Brandt et al. that if | G |= n ⩾4 k and if the degree sum of any pair of nonadjacent vertices is at least n , then G can be partitioned into k cycles. We prove that if the degree sum of any pair of nonadjacent vertices is at least n − k +1, then G can be partitioned into k subgraphs H i , 1⩽ i ⩽ k , where H i is a cycle or K 1 or K 2 , except G = C 5 and k =2.

On the decomposition dimension of trees

Abstract A decomposition F = {F 1 ,F 2 ,…,F r } of the edge set of a graph G is called a resolving r -decomposition if for any pair of edges e 1 and e 2 , there exists an index i such that d ( e 1 , F i )≠ d ( e 2 , F i ), where d ( e , F ) denotes the distance from e to F . The decomposition dimension dec( G ) of a graph G is the least integer r such that there exists a resolving r -decomposition. It is proved that for any k ⩾3 and r⩾⌈ log 2 k⌉+1 , there exists a tree T such that the maximum degree of T is k and dec( T ) is r . The relation between the decomposition dimension and the diameter of a tree is also discussed.

Independence number and vertex-disjoint cycles

In this paper we consider graphs which have no k vertex-disjoint cycles. For given integers k,@a let f(k,@a) be the maximum order of a graph G with independence number @a(G)=<@a, which has no k vertex-disjoint cycles. We prove that f(k,@a)[emailxa0protected] if 1=<@a=<5 or 1=[emailxa0protected] in general. We also prove the following results: (1) there exists a constant c@a (depending only on @a) such that f(k,@a)=<3k+c@a, (2) there exists a constant tk (depending only on k) such that f(k,@a)=<[emailxa0protected]+tk, and (3) there exists no absolute constant c such that f(k,@a)=

Long cycles in triangle-free graphs with prescribed independence number and connectivity

The Chvatal-Erdos theorem says that a 2-connected graph with α(G)≤κ(G) is hamiltonian. We extend this theorem for triangle-free graphs. We prove that if G is a 2-connected triangle-free graph of order n with α(G)≤2κ(G) - 2, then every longest cycle in G is dominating, and G has a cycle of length at least min{n - α(G) + κ(G), n}.

Ore-type degree condition for heavy paths in weighted graphs

A weighted graph is one in which every edge e is assigned a nonnegative number w(e), called the weight of e. For a vertex v of a weighted graph, d^w(v) is the sum of the weights of the edges incident to v. And the weight of a path is the sum of the weights of the edges belonging to it. In this paper, we give a sufficient condition for a weighted graph to have a heavy path which joins two specified vertices. Let G be a 2-connected weighted graph and let x and y be distinct vertices of G. Suppose that d^w(u)+d^w(v)>=2d for every pair of non-adjacent vertices u and [emailxa0protected]?V(G)@?{x,y}. Then x and y are joined by a path of weight at least d, or they are joined by a Hamilton path. Also, we consider the case when G has some vertices whose weighted degree are not assumed.