John R. Schmitt

Middlebury College · Department of Mathematics

INTEGERS: 22 (2022) A102

Label the vertices of the complete graph Kv with the integers {0, 1, …, v − 1} and define the length of the edge between the vertices x and y to be min (|x − y|,v − |x − y|). Let L be a multiset of size v − 1 with underlying set contained in {1, …, ⌊v/2⌋}. The Buratti-Horak-Rosa Conjecture is that there is a Hamiltonian path in Kv whose edge length...

The conjecture, still widely open, posed by Marco Buratti, Peter Horak and Alex Rosa states that a list L of v−1 positive integers not exceeding ⌊v2⌋ is the list of edge-lengths of a suitable Hamiltonian path of the complete graph with vertex-set {0,1,…,v−1} if and only if, for every divisor d of v, the number of multiples of d appearing in L is at...

Label the vertices of the complete graph $K_v$ with the integers $\{ 0, 1, \ldots, v-1 \}$ and define the length of the edge between $x$ and $y$ to be $\min( |x-y| , v - |x-y| )$. Let $L$ be a multiset of size $v-1$ with underlying set contained in $\{ 1, \ldots, \lfloor v/2 \rfloor \}$. The Buratti-Horak-Rosa Conjecture is that there is a Hamilton...

We generalize the notion of Davenport constants to a `higher degree' and obtain various lower and upper bounds, which are sometimes exact as is the case for certain finite commutative rings of prime power cardinality. Two simple examples that capture the essence of these higher degree Davenport constants are the following. 1) Suppose $n = 2^k$, the...

Let be an abelian group and consider a subset with . Given an ordering of the elements of , define its partial sums by and for . We consider the following conjecture of Alspach: for any cyclic group and any subset with , it is possible to find an ordering of the elements of such that no two of its partial sums and are equal for . We show that Alspa...

Let $(G,+)$ be an abelian group and consider a subset $A \subseteq G$ with $|A|=k$. Given an ordering $(a_1, \ldots, a_k)$ of the elements of $A$, define its {\em partial sums} by $s_0 = 0$ and $s_j = \sum_{i=1}^j a_i$ for $1 \leq j \leq k$. We consider the following conjecture of Alspach: For any cyclic group $\Z_n$ and any subset $A \subseteq \Z_...

A 1993 result of Alon and Füredi gives a sharp upper bound on the number of zeros of a multivariate polynomial in a finite grid over an integral domain. We give a generalization of the Alon-Füredi Theorem and discuss the relationship between Alon-Füredi, our generalization and the results of DeMillo-Lipton, Schwartz and Zippel. A direct coding theo...

We present a restricted variable generalization of Warning’s Second Theorem (a result giving a lower bound on the number of solutions of a low degree polynomial system over a finite field, assuming one solution exists). This is analogous to Schauz-Brink’s restricted variable generalization of Chevalley’s Theorem (a result giving conditions for a lo...

A 1993 result of Alon and F\"uredi gives a sharp upper bound on the number of zeros of a multivariate polynomial over an integral domain in a finite grid in terms of the degree of the polynomial. This result was recently generalized to polynomials over an arbitrary commutative ring, assuming a certain "Condition (D)" on the grid which holds vacuous...

We present a restricted variable generalization of Warning's Second Theorem (a result giving a lower bound on the number of solutions of a low degree polynomial system over a finite field, assuming one solution exists). This is analogous to Brink's restricted variable generalization of Chevalley's Theorem (a result giving conditions for a low degre...

In Martin Gardner's October 1976 Mathematical Games column in Scientific American, he posed the following problem: "What is the smallest number of [queens] you can put on an [n x n chessboard] such that no [queen] can be added without creating three in a row, a column, or, except in the case when n is congruent to 3 modulo 4, in which case one less...

We investigate the maximum number of edges in a graph with a prescribed number of 1-factors. We also examine the structure of such extremal graphs.

Given a family of (hyper)graphs F a (hyper)graph G is said to be F-saturated if G is F-free for any F ∈ F but for any edge e in the complement of G the (hyper)graph G + e contains some F ∈ F. We survey the problem of determining the minimum size of an F-saturated (hyper)graph and collect many open problems and conjectures.

Given a family of graphs ℱ, a graph G is ℱ-saturated if no element of ℱ is a subgraph of G, but for any edge e in G ¯, some element of ℱ is a subgraph of G+e. Let sat(n,ℱ) denote the minimum number of edges in an ℱ-saturated graph of order n. For graphs G, H 1 ,⋯,H k , we write that G→(H 1 ,⋯,H k ) if every k-coloring of E(G) contains a monochromat...

We consider a variation of the classical Turan-type extremal prob- lem as introduced by Erdýos et al. in (7). Letbe an n-element graphic se- quence, and �(�) be the sum of the terms in �, that is the degree sum. Let H be a graph. We wish to determine the smallest m such that any n-term graphic sequencehaving �(�) � m has some realization containing...

We extend the notion of a potentially H-graphic sequence as follows. Let A and B be nonnegative integer sequences. The sequence pair S = (A,B) is said to be bigraphic if there is some bipartite graph G = (X ∪ Y,E) such that A and B are the degrees of the vertices in X and Y , respectively. If S is a bigraphic pair, let σ(S) denote the sum of the te...

Gould, Jacobson and Lehel (Combinatorics, Graph Theory and Algorithms, Vol.I (1999) 451-460) considered a variation of the classical Turan-type extremal problems as follows: for any simple graph H, determine the smallest even integer (H,n) such that every n-term graphic sequence = (d1,d2,...,dn) with term sum ( ) = d1 +d2 +···+dn (H,n) has a realiz...

A nonincreasing sequence of nonnegative integers pi = (d(1), d(2)..... d(n)) is graphic if there is a (simple) graph G of order n having degree sequence pi. In this case, G is said to realize pi. For a given graph H, a graphic sequence pi is potentially H-graphic if there is some realization of pi containing H as a (weak) subgraph. Let sigma(pi) de...

A graph G is said to be K2;3-saturated if G contains no copy of K2;3 as a subgraph, but for any edge e in the complement of G the graph G + e does contain a copy of K2;3. The minimum number of edges of a K2;2- saturated graph of given order n was precisely determined by Ollmann in 1972. Here, we determine the asymptotic behavior for the minimum num...

Given a graph G on n vertices and a distribution, D, of pebbles on the vertices of G, we define a pebbling move to be the removal of two pebbles from a given vertex and the placement of one on an adjacent vertex. If D has n pebbles and if after a sequence of pebbling moves we can place a pebble on any specified vertex then we call G Class 0. We giv...

A graph G is said to be K 2,3-saturated if G contains no copy of K 2,3 as a subgraph, but for any edge e in the complement of G the graph G + e does contain a copy of K 2,3. The minimum number of edges of a K 2,2-saturated graph of given order n was precisely determined by Ollmann in 1972. Here, we determine the asymptotic behavior for the minimum...

For any simple graph H, let (H,n) be the minimum m so that for any realizable degree sequence = (d1,d2,... ,dn) with sum of degrees at least m, there exists an n-vertex graph G witnessing that contains H as a weak subgraph. Let Fk denote the friendship graph on 2k+1 vertices, that is, the graph of k triangles intersecting in a single vertex. In thi...

Abstract A graph G is said to be F-saturated if G does not contain a copy of F as a subgraph and G + e contains a copy of F as a subgraph for any edge e contained in the complement of G. Erdýos, Hajnal and Moon in [3] determined the minimum number of edges, sat(n,F), such that a graph G on n vertices must have when F is a t-clique. Later, Ollmann [...

We consider a variation of the classical Turán-type extremal problem as introduced by Erdős, Jacobson and Lehel in [Erdős, P., M. Jacobson, and J. Lehel, Graphs realizing the same degree sequence and their respective clique numbers, Graph Theory, Combinatorics and Applications, 1, 1991, ed. Alavi, Chartrand, Oellerman and Schwenk, 439–449]. Let π b...

Ag raphG is said to be Cl-saturated if G contains no cycle of length l, but for any edge in the complement of G the graph G + e does contain a cycle of length l. The minimum number of edges of a Cl-saturated graph was shown by Barefoot et al. to be between n + c1 n l and n + c2 n l for some positive constants c1 and c2 .T his conrmed a conjecture o...

A cycle in a multipartite graph G is gregarious if it contains at most one vertex from each partite set of G. The complete n-partite graph with partite sets of size m, denoted by K n (m) is shown to have a decomposition into gregarious 4-cycles. The notion of a gregarious 4-cycle decomposition of this type was introduced in [3]. A 4-cycle decomposi...