A graph H is an edge reconstruction of the graph G if there is a bijection p from E(G) to E(H) such that for each edge e in E(G), G\e is isomorphic to EI\fi(e). (Here G\e denotes the graph obtained from G by deleting the edge e.) We call G edge reconstructible if any edge reconstruction of G is isomorphic to G. The well-known edge-reconstruction conjecture, due to Harary [4], asserts that any g...

There is a conjecture due to Shaoji 3], about uniquely vertex r-colorable graphs which states: \ If G is a uniquely vertex r-colorable graph with order n and size (r ? 1)n ? ? r 2 , then G contains a K r as its subgraph." In this paper for any natural number r we construct a K r-free, uniquely r-colorable graph with (r ? 1)n ? ? r 2 edges. These families of graphs are indeed counter examples to...

Harary’s edge reconstruction conjecture states that a graph G=(V; E) with at least four edges is uniquely determined by the multiset of its edge-deleted subgraphs, i.e. the graphs of the form G − e for e∈E. It is well-known that this multiset uniquely determines the degree sequence of a graph with at least four edges. In this note we generalize this result by showing that the degree sequence of...

Let G be a graph of order n(G), minimum degree (G) and connectivity (G). Chartrand and Harary [Graphs with prescribed connectivities, in: P. Erdös, G. Katona (Eds.), Theory of Graphs, Academic Press, NewYork, 1968, pp. 61–63] gave the following lower bound on the connectivity (G) 2 (G)+ 2− n(G). Topp and Volkmann [Sufficient conditions for equality of connectivity and minimum degree of a graph,...

We introduce a family of multi-way Cheeger-type constants {h k , k = 1, 2, . . . , N} on a signed graph Γ = (G, σ) such that h k = 0 if and only if Γ has k balanced connected components. These constants are switching invariant and bring together in a unified viewpoint a number of important graph-theoretical concepts, including the classical Cheeger constant, the non-bipartiteness parameter of D...

We investigate families of two-dimensional simplicial complexes defined in terms of vertex decompositions. They include nonevasive complexes, strongly collapsible complexes of Barmak and Miniam and analogues of 2-trees of Harary and Palmer. We investigate the complexity of recognition problems for those families and some of their combinatorial properties. Certain results follow from analogous d...

A (k; g)-cage is a k-regular graph of girth g with minimum order. In this work, for all k ≥ 2 and g ≥ 5 odd, we present an upper bound of the order of a (k; g + 1)-cage, which depends on the order of a (k; g)-cage, improving a previous result of Sauer of 1967. We also show that every (k; 11)-cage contains a cycle of length 12, confirming a case of a conjecture of Harary and Kóvacs of 1983.

The interval number of a graph G, denoted i(G), is the least positive integer t for which G is the intersection graph of a family of sets each of which is the union of at most t cIosed intervals of the real line IR. Trotter and Harary showed that the interval number of the complete bipartite graph K(m, n) is [(mn + I)/(m + n)]. Matthews showed that the interval number of the complete multiparti...