The harmonious coloring of an undirected simple graph is a vertex coloring such that adjacent vertices are assigned different colors and each pair of colors appears together on at most one edge. The harmonious chromatic number of a graph is the least number of colors used in such a coloring. The harmonious chromatic number of a path is known, whereas the problem to find the harmonious chromatic number is NP-hard even for trees with pathwidth at most 2. Hence, we consider the harmonious coloring of trees with pathwidth 1, which are also known as caterpillars. This paper shows the harmonious chromatic number of a caterpillar with at most one vertex of degree more than 2. We also show the upper bound of the harmonious chromatic number of a 3-regular caterpillar.

Suppose that each arc in a digraph D = (V,A) has two costs of non-negative integers, called a spine cost and a leaf cost. A caterpillar is a directed tree consisting of a single directed path (of spine arcs) and leaf vertices each of which is incident to the directed path by exactly one incoming arc (leaf arc). For a given terminal set K ⊆ V, we study the problem of finding a caterpillar in D such that it contains all terminals in K and its total cost is minimized, where the cost of each arc in the caterpillar depends on whether it is used as a spine arc or a leaf arc. In this paper, we first show that the problem is NP-hard for any fixed constant number of terminals with |K| ≥ 3, while it is solvable in polynomial time for at most two terminals. We also give an inapproximability result for any fixed constant number of terminals with |K| ≥ 3. Finally, we give a linear-time algorithm to solve the problem for digraphs with bounded treewidth, where the treewidth for a digraph D is defined as the one for the underlying graph of D. Our algorithm runs in linear time even if |K| = O(|V|), and the hidden constant factor of the running time is just a single exponential of the treewidth.

The path coloring problem is to assign the minimum number of colors to a given set P of directed paths on a given symmetric digraph D so that no two paths sharing an arc have the same color. The problem has applications to efficient assignment of wavelengths to communications on WDM optical networks. In this paper, we show that the path coloring problem is NP-hard even if the underlying graph of D is restricted to a binary caterpillar. Moreover, we give a polynomial time algorithm which constructs, given a binary caterpillar G and a set P of directed paths on the symmetric digraph associated with G, a path coloring of P with at most colors, where L is the maximum number of paths sharing an edge. Furthermore, we show that no local greedy path coloring algorithm on caterpillars in general uses less than colors.

Some qualitative properties of an inductively coupled circuit containing two Josephson junction elements with a dc source are investigated. The system is described by a four–dimensional autonomous differential equation. However, the phase space can be regarded as S1×R3 because the system has a periodicity for the invariant transformation. In this paper, we study the properties of periodic solutions winding around S1 as a bifurcation problem. Firstly, we analyze equilibria in this system. The bifurcation diagram of equilibria and its topological classification are given. Secondly, the bifurcation diagram of the periodic solutions winding around S1 are calculated by using a suitable Poincar mapping, and some properties of periodic solutions are discussed. From these analyses, we clarify that a periodic solution so–called "caterpillar solution" is observed when the two Josephson junction circuits are weakly coupled.