Given a set of integers S; G(S) = (S; E) is a graph, where the edge uv exists if and only if u+ v∈ S. A graph G = (V; E) is an integral sum graph or ISG if there exists a set S ⊂ Z such that G=G(S). This set is called a labeling of G. The main results of this paper concern regular ISGs. It is proved that all 2-regular graphs with the exception of C4 are integral sum graphs and that for every po...

Let %I( R, S) denote the class of all m X n matrices of O’s and l’s having row sum vector R and column sum vector S. The interchange graph G( R, S) is the graph where the vertices are the matrices in %(R, S) and where two matrices are joined by an edge provided they differ by an interchange. We characterize those 81 (R, S) for which the graph C( R, S) has diameter at most 2 and those YI( R, S) ...

For a connected graph G of order n ≥ 3, let f : E(G) → Zn be an edge labeling of G. The vertex labeling f ′ : V (G) → Zn induced by f is defined as f (u) = ∑ v∈N(u) f(uv), where the sum is computed in Zn. If f ′ is one-to-one, then f is called a modular edge-graceful labeling and G is a modular edge-graceful graph. A modular edge-graceful labeling f of G is nowhere-zero if f(e) 6= 0 for all e ∈...

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, a vertex labeling $f : V(G)rightarrow mathbb{Z}_2$ induces an edge labeling $ f^{+} : E(G)rightarrow mathbb{Z}_2$ defined by $f^{+}(xy) = f(x) + f(y)$, for each edge $ xyin E(G)$.  For each $i in mathbb{Z}_2$, let $ v_{f}(i)=|{u in V(G) : f(u) = i}|$ and $e_{f^+}(i)=|{xyin E(G) : f^{+}(xy) = i}|$. A vertex labeling $f$ of a graph $G...

A pair of spanning trees rooted at a vertex r are independent if for every vertex v the pair of unique tree paths from v to the root r are disjoint. This paper presents the rst analysis of the path lengths involved in independent spanning trees in 2-edge-connected and 2-vertex-connected graphs. We present upper and lower bounds on the stretch factors of pairs of independent spanning trees, wher...

We introduce and study balanced online graph avoidance games on the random graph process. The game is played by a player we call Painter. Edges of the complete graph with n vertices are revealed two at a time in a random order. In each move, Painter immediately and irrevocably decides on a balanced coloring of the new edge pair: either the first edge is colored red and the second one blue or vi...

An essentially k-edge connected graph G is a connected graph such that deleting less than k edges from G cannot result in two nontrivial components. In this paper we prove that if an essentially 2-edge-connected graph G satisfies that for any pair of leaves at distance 4 in G there exists another leaf of G that has distance 2 to one of them, then the square G2 has a connected even factor with m...

If the network topology is modeled by an undirected graph and we consider link failures, then a natural graph theoretic measure of vulnerability is the size of a minimum cut, that is, the minimum number of edges that have to be removed to disconnect the graph, which is called the (edge-)connectivity of the graph. This number, at the same time, is the guaranteed number of edge disjoint paths bet...

In 1955 Kotzig proved that every planar 3-connected graph contains an edge such that sum of degrees of its endvertices is at most 13. Moreover, if the graph does not contain 3-vertices, then this sum is at most 11. Such an edge is called light. The well-known result of Steinitz that the 3-connected planar graphs are precisely the skeletons of 3-polytopes, gives an additional trump to Kotzig’s t...

1. INTRODUCTION. A Hamiltonian cycle in a graph is a path in the graph which visits each vertex exactly once and returns to the starting vertex. Let í µí°¾ í µí±› be a weighted complete graph with í µí±› vertices. We define the weight of an edge as the square of the distance between two end points of the edge. The weight of a path í µí±ƒ is the sum of the weights of all edges in the path and de...