International Journal of Mathematics and Mathematical Sciences, 1989
Let E be a topological vector space of scalar sequences, with topologyτ;(E,τ)satisfies the closed... more Let E be a topological vector space of scalar sequences, with topologyτ;(E,τ)satisfies the closed neighborhood condition iff there is a basis of neighborhoods at the origin, forτ, consisting of sets whlch are closed with respect to the topologyπof coordinate-wise convergence on E;(E,τ)satisfies the filter condition iff every filter, Cauchy with respect toτ, convergent with respect toπ, converges with respect toτ.Examples are given of solid (definition below) normed spaces of sequences which (a) fail to satisfy the filter condition, or (b) satisfy the filter condition, but not the closed neighborhood condition. (Robertson and others have given examples fulfilling (a), and examples fulfilling (b), but these examples were not solid, normed sequence spaces.) However, it is shown that among separated, separable solid pairs(E,τ), the filter and closed neighborhood conditions are equivalent, and equivalent to the usual coordinate sequences constituting an unconditional Schauder basis for(E...
The main theorem gives a class of inequalities concerning finite hypergraphs with a fixed number ... more The main theorem gives a class of inequalities concerning finite hypergraphs with a fixed number of vertices per edge. One corollary of the general result: if there are edges, then there is an edge such that the geometric mean of the degrees of its vertices is greater than or equal to the arithmetic mean degree of all the vertices. If the hypergraph is not regular, the inequality is strict. Another corollary of the main result is used to derive an inequality for simple graphs relating the number of edges, the number of vertices, and the largest number of triangles based on a fixed edge in the graph. Finally, some results are derived relating the degrees of non-adjacent vertices of a simple graph to the average degree.
The graphs with Hall number at most 2 form a class of graphs within which the chromatic number eq... more The graphs with Hall number at most 2 form a class of graphs within which the chromatic number equals the choice (list-chromatic) number. This class has a forbidden-induced-subgraph characterization which has not yet been found, although a fairly imposing collection of minimal forbidden induced subgraphs has been assembled. In this paper we add to the collection, most notably adding (i) K5 with an ear of length 2 attached; (ii) K4 with an ear of any length> 2 attached; (iii) any cycle together with "two triangles based on incident edges on the cycle; (iv) any odd cycle together with two triangles based on non-incident edges of the cycle; and (v) any even cycle together with three triangles based on non-incident edges of the cycle.
We continue the classification of the regular simple graphs in which, for some t, any two adjacen... more We continue the classification of the regular simple graphs in which, for some t, any two adjacent vertices have exactly t common neighbors, and the union of their neighbor sets misses exactly two vertices. Previously it was shown that for any such graph with n vertices, if t > 0 then t + 8 ≤ n ≤ 3t + 6. Here we show that there is exactly one such graph on n = 3t + 6 vertices, for each t = 1, 2,. .. , namely K t+2,t+2,t+2 minus a two-factor consisting of triangles. Let G be a simple graph. For an edge e ∈ E(G) with end-vertices u, v let t(e) = |N(u) ∩ N(v)| and let J(e) = |N(u) ∪ N(v)| (with neighborhoods taken in G, of course). Let t(G) = |E(G)| −1 e∈E(G) t(e), and let J(G) = max e∈E(G) J(e). It is shown in [3] that if G has m edges and n vertices, then 4m ≤ n(J(G) + t(G)), with equality if and only if G is regular and t is a constant function (equivalently, G is regular and J is a constant function; observe that J(e) + t(e) = d(u) + d(v)). This conclusion also holds if J(G) is defined to be an arithmetic mean, and t(G) is a max. Clearly this result generalizes Mantel's famous theorem, i.e. Turan's theorem with r = 2 (see [5]). To agree with the notation in [3] and [4], let us denote by ET (n, J, t) the set of extremal graphs for the inequality above, on n vertices with J = J(G) and t = t(G).
An edge-regular graph is a regular graph in which, for some $\lambda$, any two adjacent vertices ... more An edge-regular graph is a regular graph in which, for some $\lambda$, any two adjacent vertices have exactly $\lambda$ common neighbors. This paper is about the existence and structure of edge-regular graphs with $\lambda =1$ and about edge-regular graphs with $\lambda >1$ which have local neighborhood structure analogous to that of the edge-regular graphs with $\lambda =1$.
If you treasure semantic precision, you might name the volcano that erupted (figuratively speakin... more If you treasure semantic precision, you might name the volcano that erupted (figuratively speaking!) in 1973: Coloring Problems in Geometrically Defined Hypergraphs.
Page 1. Darrel Hankerson Greg A. Harris Peter D. Johnson, Jr. Introduction to &am... more Page 1. Darrel Hankerson Greg A. Harris Peter D. Johnson, Jr. Introduction to < Information Theory and Data Compression Second Edition Page 2. Page 3. Page 4. Page 5. Introduction to Information Theory and Data Compression Second Edition This One DOE3-9KT-6615 ...
... of [1]. The Babai numbers were also discovered independently by Paul Erdös in [8]; the defini... more ... of [1]. The Babai numbers were also discovered independently by Paul Erdös in [8]; the definition is confined to the case XDR2, but it ... the 20-year-old Francis Guthrie started a mathe-matical avalanche by posing the four-color conjecture to his brother Frederick, who transmitted ...
In an ordinary list multicoloring of a graph, the vertices are "colored" with subsets of pre-assi... more In an ordinary list multicoloring of a graph, the vertices are "colored" with subsets of pre-assigned finite sets (called "lists") in such a way that adjacent vertices are colored with disjoint sets. Here we consider the analog of such colorings in which the lists are measurable sets from an arbitrary atomless, semifinite measure space, and the color sets are to have prescribed measures rather than prescribed cardinalities. We adapt a proof technique of Bollobás and Varopoulos to prove an analog of one of the major theorems about ordinary list multicolorings, a generalization and extension of Hall's marriage theorem, due to Cropper, Gyárfás, and Lehel.
Let P denote the set of positive integers. For m, n ∈ P, n ⩾ 2, a Kraft sequence of length m for ... more Let P denote the set of positive integers. For m, n ∈ P, n ⩾ 2, a Kraft sequence of length m for alphabet size n is a sequence (l1,...,lm) of the lengths of the words in a prefix-free list of words formed from an alphabet of n letters. For instance, because none of the binary words ...
International Journal of Mathematics and Mathematical Sciences, 2005
In 1986, Johnson and Perry proved a class of inequalities for uniform hypergraphs which included ... more In 1986, Johnson and Perry proved a class of inequalities for uniform hypergraphs which included the following: for any such hypergraph, the geometric mean over the hyperedges of the geometric means of the degrees of the nodes on the hyperedge is no less than the average degree in the hypergraph, with equality only if the hypergraph is regular. Here, we prove a wider class of inequalities in a wider context, that of edge-weighted uniform hypergraphs.
International Journal of Mathematics and Mathematical Sciences, 1989
Let E be a topological vector space of scalar sequences, with topologyτ;(E,τ)satisfies the closed... more Let E be a topological vector space of scalar sequences, with topologyτ;(E,τ)satisfies the closed neighborhood condition iff there is a basis of neighborhoods at the origin, forτ, consisting of sets whlch are closed with respect to the topologyπof coordinate-wise convergence on E;(E,τ)satisfies the filter condition iff every filter, Cauchy with respect toτ, convergent with respect toπ, converges with respect toτ.Examples are given of solid (definition below) normed spaces of sequences which (a) fail to satisfy the filter condition, or (b) satisfy the filter condition, but not the closed neighborhood condition. (Robertson and others have given examples fulfilling (a), and examples fulfilling (b), but these examples were not solid, normed sequence spaces.) However, it is shown that among separated, separable solid pairs(E,τ), the filter and closed neighborhood conditions are equivalent, and equivalent to the usual coordinate sequences constituting an unconditional Schauder basis for(E...
The main theorem gives a class of inequalities concerning finite hypergraphs with a fixed number ... more The main theorem gives a class of inequalities concerning finite hypergraphs with a fixed number of vertices per edge. One corollary of the general result: if there are edges, then there is an edge such that the geometric mean of the degrees of its vertices is greater than or equal to the arithmetic mean degree of all the vertices. If the hypergraph is not regular, the inequality is strict. Another corollary of the main result is used to derive an inequality for simple graphs relating the number of edges, the number of vertices, and the largest number of triangles based on a fixed edge in the graph. Finally, some results are derived relating the degrees of non-adjacent vertices of a simple graph to the average degree.
The graphs with Hall number at most 2 form a class of graphs within which the chromatic number eq... more The graphs with Hall number at most 2 form a class of graphs within which the chromatic number equals the choice (list-chromatic) number. This class has a forbidden-induced-subgraph characterization which has not yet been found, although a fairly imposing collection of minimal forbidden induced subgraphs has been assembled. In this paper we add to the collection, most notably adding (i) K5 with an ear of length 2 attached; (ii) K4 with an ear of any length> 2 attached; (iii) any cycle together with "two triangles based on incident edges on the cycle; (iv) any odd cycle together with two triangles based on non-incident edges of the cycle; and (v) any even cycle together with three triangles based on non-incident edges of the cycle.
We continue the classification of the regular simple graphs in which, for some t, any two adjacen... more We continue the classification of the regular simple graphs in which, for some t, any two adjacent vertices have exactly t common neighbors, and the union of their neighbor sets misses exactly two vertices. Previously it was shown that for any such graph with n vertices, if t > 0 then t + 8 ≤ n ≤ 3t + 6. Here we show that there is exactly one such graph on n = 3t + 6 vertices, for each t = 1, 2,. .. , namely K t+2,t+2,t+2 minus a two-factor consisting of triangles. Let G be a simple graph. For an edge e ∈ E(G) with end-vertices u, v let t(e) = |N(u) ∩ N(v)| and let J(e) = |N(u) ∪ N(v)| (with neighborhoods taken in G, of course). Let t(G) = |E(G)| −1 e∈E(G) t(e), and let J(G) = max e∈E(G) J(e). It is shown in [3] that if G has m edges and n vertices, then 4m ≤ n(J(G) + t(G)), with equality if and only if G is regular and t is a constant function (equivalently, G is regular and J is a constant function; observe that J(e) + t(e) = d(u) + d(v)). This conclusion also holds if J(G) is defined to be an arithmetic mean, and t(G) is a max. Clearly this result generalizes Mantel's famous theorem, i.e. Turan's theorem with r = 2 (see [5]). To agree with the notation in [3] and [4], let us denote by ET (n, J, t) the set of extremal graphs for the inequality above, on n vertices with J = J(G) and t = t(G).
An edge-regular graph is a regular graph in which, for some $\lambda$, any two adjacent vertices ... more An edge-regular graph is a regular graph in which, for some $\lambda$, any two adjacent vertices have exactly $\lambda$ common neighbors. This paper is about the existence and structure of edge-regular graphs with $\lambda =1$ and about edge-regular graphs with $\lambda >1$ which have local neighborhood structure analogous to that of the edge-regular graphs with $\lambda =1$.
If you treasure semantic precision, you might name the volcano that erupted (figuratively speakin... more If you treasure semantic precision, you might name the volcano that erupted (figuratively speaking!) in 1973: Coloring Problems in Geometrically Defined Hypergraphs.
Page 1. Darrel Hankerson Greg A. Harris Peter D. Johnson, Jr. Introduction to &am... more Page 1. Darrel Hankerson Greg A. Harris Peter D. Johnson, Jr. Introduction to < Information Theory and Data Compression Second Edition Page 2. Page 3. Page 4. Page 5. Introduction to Information Theory and Data Compression Second Edition This One DOE3-9KT-6615 ...
... of [1]. The Babai numbers were also discovered independently by Paul Erdös in [8]; the defini... more ... of [1]. The Babai numbers were also discovered independently by Paul Erdös in [8]; the definition is confined to the case XDR2, but it ... the 20-year-old Francis Guthrie started a mathe-matical avalanche by posing the four-color conjecture to his brother Frederick, who transmitted ...
In an ordinary list multicoloring of a graph, the vertices are "colored" with subsets of pre-assi... more In an ordinary list multicoloring of a graph, the vertices are "colored" with subsets of pre-assigned finite sets (called "lists") in such a way that adjacent vertices are colored with disjoint sets. Here we consider the analog of such colorings in which the lists are measurable sets from an arbitrary atomless, semifinite measure space, and the color sets are to have prescribed measures rather than prescribed cardinalities. We adapt a proof technique of Bollobás and Varopoulos to prove an analog of one of the major theorems about ordinary list multicolorings, a generalization and extension of Hall's marriage theorem, due to Cropper, Gyárfás, and Lehel.
Let P denote the set of positive integers. For m, n ∈ P, n ⩾ 2, a Kraft sequence of length m for ... more Let P denote the set of positive integers. For m, n ∈ P, n ⩾ 2, a Kraft sequence of length m for alphabet size n is a sequence (l1,...,lm) of the lengths of the words in a prefix-free list of words formed from an alphabet of n letters. For instance, because none of the binary words ...
International Journal of Mathematics and Mathematical Sciences, 2005
In 1986, Johnson and Perry proved a class of inequalities for uniform hypergraphs which included ... more In 1986, Johnson and Perry proved a class of inequalities for uniform hypergraphs which included the following: for any such hypergraph, the geometric mean over the hyperedges of the geometric means of the degrees of the nodes on the hyperedge is no less than the average degree in the hypergraph, with equality only if the hypergraph is regular. Here, we prove a wider class of inequalities in a wider context, that of edge-weighted uniform hypergraphs.
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