Severino V. Gervacio

Disjoint subsets of integers having a constant sum

Abstract We prove that for positive integers n, m and k, the set {1, 2,…,n} of integers contains k disjoint subsets having a constant sum m if and only if 2k − 1;⩽m⩽n(n+1) (2k)

Score sequences: lexicographic enumeration and tournament construction

In this paper we give an algorithm for generating all tournament score sequences of a given length in lexicographic order. We also show how to construct a tournament with a given score sequence.

Resistance distance in complete n -partite graphs

We may view any graph as a network of resistors each having a resistance of 1 ? . The resistance distance between a pair of vertices in a graph is defined as the effective resistance between the two vertices. This function is known to be a metric on the vertex-set of any graph. The main result of this paper is an explicit expression for the resistance distance between any pair of vertices in the complete n -partite graph K m 1 , m 2 , ? , m n .

The bandwidth of a tree with k leaves is at most k 2

Abstract The bandwidth B(G) of a finite simple graph G is the minimum of the quantity max {ƒ(x)-ƒ(y) : xy ∈ E(G)} taken over all injective integer labellings ƒ of G. We prove that if a tree T has k leaves then B(T) ⩽ [ k 2 ] . This improves the previously known upper bound B(T) ⩽ ∥ V(T)∥/2.

Planar unit-distance graphs having planar unit-distance complement

A graph G=(V,E) is called a unit-distance graph in the plane if there is an embedding of V into the plane such that every pair of adjacent vertices are at unit distance apart. If an embedding of V satisfies the condition that two vertices are adjacent if and only if they are at unit distance apart, then G is called a strict unit-distance graph in the plane. A graph G is a (strict) co-unit-distance graph, if both G and its complement are (strict) unit-distance graphs in the plane. We show by an exhaustive enumeration that there are exactly 69 co-unit-distance graphs (65 are strict co-unit-distance graphs), 55 of which are connected (51 are connected strict co-unit-distance graphs), and seven are self-complementary.

A Note on Lights-Out-Puzzle: Parity-State Graphs

A state of a graph G is an assignment of 0 or 1 to each vertex of G. A move of a state consists of choosing a vertex and then switching the value of the vertex as well as those of its neighbors. Two states are said to be equivalent if one state can be changed to the other by a series of moves. A parity-state graph is defined to be a graph in which two states are equivalent if and only if the numbers of 1’s in the two states have the same parity. We characterize parity-state graphs and present some constructions of parity-state graphs together with applications. Among other things, it is proved that the one-skeleton of the 3-polytope obtained from a simple 3-polytope by cutting off all vertices is a parity-state graph.

Subdividing a Graph Toward a Unit-distance Graph in the Plane

The subdivision number of a graph G is defined to be the minimum number of extra vertices inserted into the edges of G to make it isomorphic to a unit-distance graph in the plane. Lett (n) denote the maximum number of edges of a C4-free graph on n vertices. It is proved that the subdivision number of Knlies betweenn (n? 1)/2 ?t(n) and (n? 2)(n? 3)/2 + 2, and that of K(m, n) equals (m? 1)(n?m) forn?m(m? 1).

Folding Wheels and Fans

Abstract. If two non-adjacent vertices of a connected graph that have a common neighbor are identified and the resulting multiple edges are reduced to simple edges, then we obtain another graph of order one less than that of the original graph. This process can be repeated until the resulting graph is complete. We say that we have folded the graph onto complete graph. This process of folding a connected graph G onto a complete graph induces in a very natural way a partition of the vertex-set of G. We denote by F(G) the set of all complete graphs onto which G can be folded. We show here that if p and q are the largest and smallest orders, respectively, of the complete graph in F(Wn) or F(Fn), then Ks is in F(Wn) or F(Fn) for each s, q≤s≤p. Lastly, we shall also determine the exact values of p and q.

Trees with diameter less than 5 and non-singular complement

A graph G is said to be singular if its adjacency matrix is singular; otherwise it is said to be non-singular. Non-singular trees have been completely characterized. Here we investigate the complement of a tree with diameter less than 5 for singularity or non-singularity. It is easy to see that every tree with diameter d ⩽ 2 has a singular complement. We prove that the complement of any tree with diameter 3 is non-singular. We prove also that if T is a tree with diameter 4 and central vertex xo, then the complement T of T is non-singular if and only if T/x0 contains at most one component P2. Given any tree T, we prove that T is singular if T⋎ has at least two components P2 for some vertex v ϵ T.

Solvable Trees

A state of a simple graph G is an assignment of either a 0 or 1 to each of its vertices. For each vertex i of G , we define the move [i ] to be the switching of the state of vertex i , and each neighbor of i , from 0 to 1, or from 1 to 0. The given initial state of G is said to be solvable if a sequence of moves exists such that this state is transformed into the 0-state (all vertices have state 0.) If every initial state of G is solvable, we call G a solvable graph . We shall characterize here the solvable trees.