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...

Methodological challenges arise when one uses various Veterans Health Administration (VHA) data sources, each created for distinct purposes, to characterize length of stay (LOS). To illustrate this issue, we examined how algorithm choice affects conclusions about mental health condition (MHC)-related differences in LOS for VHA patients with diabetes nationally (n = 784,321). We assembled a reco...

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...