Michael Jacobson

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• Ph. D. Mathematics
• Professor at University of Colorado

This poster focuses on the experiences of TA Coaches in a comprehensive graduate teaching assistant training program in mathematical sciences that was designed and refined at one institution and is being replicated at two peer institutions. During program development, TA coaches were tasked with working with GTAs teaching recitation sections of col...

Peer mentoring programs are one approach to improving the pedagogical development of mathematical sciences graduate students. This paper describes the peer mentoring experiences at three institutions that have implemented a multi-faceted GTA professional development program. Data was collected from surveys and focus groups conducted with graduate t...

Peer mentoring programs are one approach to improving the pedagogical development of mathematical sciences graduate students. This paper describes the peer mentoring experiences at three institutions that have implemented a multi-faceted GTA professional development program. Data was collected from surveys and focus groups conducted with graduate t...

A graph G is H-saturated if H is not a subgraph of G but the addition of any edge from the complement of G to G results in a copy of H. The minimum number of edges (the size) of an H-saturated graph on n vertices is denoted sat(n, H), while the maximum size is the well studied extremal number, ex(n, H). The saturation spectrum for a graph H is the...

A graph G is H-saturated if H is not a subgraph of G but the addition of any edge from G to G results in a copy of H. The minimum size of an H-saturated graph on n vertices is denoted sat(n,H), while the maximum size is the well studied extremal number, ex(n,H). The saturation spectrum for a graph H is the set of sizes of H saturated graphs between...

The need for a more robust, well-trained STEM workforce is becoming increasingly acute in the U.S., and there is a clear need to recruit and retain a larger and more diverse population of undergraduate STEM majors. While numerous efforts to improve engagement and support in the traditional P-16 classroom have been implemented successfully, it is al...

Mathematics is a central part of all STEM disciplines, and undergraduate success in mathematics courses is an increasingly critical piece of the growing national need to train the next generation of U.S. STEM professionals. Despite this, early undergraduate mathematics courses are often gatekeepers that prevent students from reaching their goals of...

A dominating path in a graph is a path P such that every vertex outside P has a neighbor on P. A result of Broersma from 1988 implies that if G is an n-vertex k-connected graph and , then G contains a dominating path. We prove the following results. The lengths of dominating paths include all values from the shortest up to at least . For , where a...

A vertex dominating path in a graph is a path P such that every vertex outside P has a neighbor on P. In 1988 H. Broersma [5] stated a result implying that every n‐vertex k‐connected graph G such that contains a vertex dominating path. We provide a short, self‐contained proof of this result and further show that every n‐vertex k‐connected graph suc...

A broom is a tree obtained by subdividing one edge of the star an arbitrary number of times. In (E. Flandrin, T. Kaiser, R. Kužel, H. Li and Z. Ryjáček, Neighborhood Unions and Extremal Spanning Trees, Discrete Math 308 (2008), 2343–2350) Flandrin et al. posed the problem of determining degree conditions that ensure a connected graph G contains a s...

Let $G$ be a fixed graph and let ${\mathcal F}$ be a family of graphs. A subgraph $J$ of $G$ is ${\mathcal F}$-saturated if no member of ${\mathcal F}$ is a subgraph of $J$, but for any edge $e$ in $E(G)-E(J)$, some element of ${\mathcal F}$ is a subgraph of $J+e$. We let $\text{ex}({\mathcal F},G)$ and $\text{sat}({\mathcal F},G)$ denote the maxim...

A (finite) sequence of nonnegative integers is graphic if it is the degree sequence of some simple graph G. Given graphs G 1 and G 2, we consider the smallest integer k such that for every k-term graphic sequence π, there is some graph G with degree sequence π with ${G_1 \subseteq G}$ or with ${G_2 \subseteq \overline{G}}$ . When the phrase “so...

Given a (multi)digraph H , a digraph D is H - linked if every injective function ι: V(H) → V(D) can be extended to an H -subdivision. In this paper, we give sharp degree conditions that ensure a sufficiently large digraph D is H -linked for arbitrary H . The notion of an H -linked digraph extends the classes of m -linked, m -ordered and strongly m...

For a fixed graph F , a graph G is F -saturated if there is no copy of F in G, but for any edge e ∉ G, there is a copy of F in G + e. The minimum number of edges in an F -saturated graph of order n will be denoted by sat(n, F ). A graph G is weakly F -saturated if there is an ordering of the missing edges of G so that if they are added one at a tim...

For s≥3s≥3 a graph is K1,sK1,s-free if it does not contain an induced subgraph isomorphic to K1,sK1,s. Cycles in K1,3K1,3-free graphs, called claw-free graphs, have been well studied. In this paper we extend results on disjoint cycles in claw-free graphs satisfying certain minimum degree conditions to K1,sK1,s-free graphs, normally called generaliz...

Let D be a directed graph of order n. An anti-directed (hamiltonian) cycle H in D is a (hamiltonian) cycle in the graph underlying D such that no pair of consecutive arcs in H form a directed path in D. In this paper we give sufficient conditions for the existence of anti-directed hamiltonian cycles. Specifically, we prove that a directed graph D o...

For a family F of graphs, a graph G is F‐saturated if G contains no member of F as a subgraph, but for any edge uv in G¯, G+uv contains some member of F as a subgraph. The minimum number of edges in an F‐saturated graph of order n is denoted sat(n,F). A subdivision of a graph H, or an H‐subdivision, is a graph G obtained from H by replacing the edg...

For a fixed (multi)graph H, a graph G is H-linked if any injection f: V(H)→V(G) can be extended to an H-subdivision in G. The notion of an H -linked graph encompasses several familiar graph classes, including k-linked, k-ordered and k-connected graphs. In this article, we give two sharp Ore-type degree sum conditions that assure a graph G is H -lin...

An n-tuple π (not necessarily monotone) is graphic if there is a simple graph G with vertex set {v1, …, vn} in which the degree of vi is the ith entry of π. Graphic n-tuples (d, …, d) and (d, …, d) pack if there are edge-disjoint n-vertex graphs G1 and G2 such that d(vi) = d and d(vi) = d for all i. We prove that graphic n-tuples π1 and π2 pack if...

For graphs G and H, H is said to be G-saturated if it does not contain a subgraph isomorphic to G, but for any edge e∈H c , the complement of H, H+e contains a subgraph isomorphic to G. The minimum number of edges in a G-saturated graph on n vertices is denoted sat(n,G). While digraph saturation has been considered with the allowance of multiple ar...

In 1963, Moon and Moser gave a bipartite analogue to Ore’s famed theorem on hamiltonian graphs. While the sharpness examples of Ore’s Theorem have been independently characterized in at least four different papers, no similar characterization exists for the Moon–Moser Theorem. In this note, we give such a characterization, consisting of one infinit...

Let G be a graph of order n and 3≤t≤n/4 be an integer. Recently, Kaneko and Yoshimoto [J Combin Theory Ser B 81(1) (2001), 100–109] provided a sharp δ(G) condition such that for any set X of t vertices, G contains a hamiltonian cycle H so that the distance along H between any two vertices of X is at least n/2t. In this article, minimum degree and c...

A spanning cycle in a graph G is called a hamiltonian cycle, and if such a cycle exists G is said to be hamiltonian. Let G be a graph and H be a subgraph of G. If G contains a hamiltonian cycle C such that E(C) \ E(H) is empty, we say that C is an H-avoiding hamiltonian cycle. Let F be any graph. If G contains an H-avoiding hamiltonian cycle for ev...

A graph of order n is said to be pancyclic if it contains cycles of all lengths from three to n. Let G be a hamiltonian graph and let x and y be vertices of G that are consecutive on some hamiltonian cycle in G. Hakimi and Schmeichel showed (2) that if d(x) + d(y) ≥ n then either G is pancyclic, G has cycles of all lengths except n − 1 or G is isom...

Let G be a graph and ℱ be a family of graphs. We say that G is ℱ-saturated if G does not contain a copy of any member of ℱ, but for any pair of nonadjacent vertices x and y in G,G+xy contains a copy of some H∈ℱ. A great deal of study has been devoted to the maximum and the minimum number of edges in an ℱ-saturated graph. Little is known, however, a...

Given a graph G, we consider the problem of finding the minimum number n such that any k edge colored complete graph on n vertices contains either a three colored triangle or a monochromatic copy of the graph G. This number is found precisely for a C 4 and all trees on at most 6 vertices and bounds are provided for general paths.

The Chvátal-Erdös theorems imply that if G is a graph of order n ≥ 3 with k(G) ≥α(G), then G is hamiltonian, and if k(G) > α(G), then G is hamiltonian-connected. We generalize these results by replacing the connectivity and independence number conditions with a weakerminimum degree and independence number condition in the presence of sufficient con...

The Chvátal-Erdős theorems imply that if G is a graph of order n≥3 with κ(G)≥α(G), then G is Hamiltonian, and if κ(G)>α(G), then G is Hamiltonian-connected. We generalize these results by replacing the connectivity and independence number conditions with a weaker minimum degree and independence number condition in the presence of sufficient connect...

A graph G is H-saturated if G does not contain H as a subgraph but for any nonadjacent vertices u and v, G + uv contains H as a subgraph. The parameter sat(H,n) is the minimum number of edges in an H-saturated graph of order n. In this paper, we determine sat(H,n) for suciently large n when H is a union of cliques of the same order, an arbitrary un...

A tournament is an oriented complete graph, and one containing no directed cycles is called transitive. A tournament T=(V, A) is called m-partition transitive if there is a partition \documentclass{article}\usepackage{amssymb} \usepackage{amsbsy}\usepackage[mathscr]{euscript}\footskip=0pc\pagestyle{empty} \begin{document}${{V}}={{X}}_{{{1}}} {\cup\...

For a fixed graph H, a graph G is H-saturated if there is no copy of H in G, but for any edge e 6∈ G, there is a copy of H in G + e. The collection of H- saturated graphs of order n is denoted by SAT(n, H), and the saturation number, sat(n, H), is the minimum number of edges in a graph in SAT(n, H). Let Tk be a tree on k vertices. The saturation nu...

Given integers k,s,t with 0≤s≤t and k≥0, a (k,t,s)-linear forest F is a graph that is the vertex disjoint union of t paths with a total of k edges and with s of the paths being single vertices. If the number of single vertex paths is not critical, the forest F will simply be called a (k,t)-linear forest. A graph G of order n≥k+t is (k,t)-hamiltonia...

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...

We prove that any cycle

A tournament T = (V, A) is arc-traceable if for each arc xy ∈ A, xy lies on a directed path containing all the vertices of V, i.e., a hamiltonian path. In this paper we give two extremal results related to arc-traceability in tournaments. First, we show that a non-arc-traceable tournament T which is m-arc-strong must have at least 2 m+1 + 4m-3 vert...

In this paper we consider the question of determining the maximum number of edges in a Hamiltonian graph of order n that contains no 2-factor with more than one cycle, that is, 2-factor Hamiltonian graphs. We obtain exact results for both bipartite graphs, and general graphs, and construct extremal graphs in each case.

We examine several extremal problems for graphs satisfying the property |N(x) ∪ N(y)| ⩾ s for every pair of nonadjacent vertices x, y ϵ V(G). In particular, values for s are found that ensure that the graph contains an s-matching, a 1-factor, a path of a specific length, or a cycle of a specific length.

Chvátal established that r(Tm, Kn) = (m – 1)(n – 1) + 1, where Tm is an arbitrary tree of order m and Kn is the complete graph of order n. This result was extended by Chartrand, Gould, and Polimeni who showed Kn could be replaced by a graph with clique number n and order n + 1 provided n ≧ 3 and m ≧ 3. We further extend these results to show that K...

Various Harniltonian-like properties are investigated in the squares of connected graphs free of some set of forbidden subgraphs. The star K1,4 the subdivision graph of K1,3, and the subdivision graph of K1,3 minus an endvertex play central roles. In particular, we show that connected graphs free of the subdivision graph of K1,3 minus an endvertex...

Bounds are determined for the Ramsey number of the union of graphs versus a fixed graph H , based on the Ramsey number of the components versus H . For certain unions of graphs, the exact Ramsey number is determined. From these formulas, some new Ramsey numbers are indicated. In particular, if magnified image . Where k i is the number of components...

A digraph D = (V, A) is arc-traceable if for each arc xy in A, xy lies on a directed path containing all the vertices of V, that is, a hamiltonian path. Given a tournament T, it is well known that it contains a directed hamiltonian path. In this article, we develop the structure necessary for a tournament T to contain an arc xy that is not on any h...

A Θ-cycle of a hypergraph is a cycle including an edge that contains at least three base points of the cycle. We show that if a hypergraph H=(V,E) has no Θ-cycle, and |e|⩾3, for every edge e∈E, then ∑e∈E(|e|-1)⩽2|V|-2 with equality if and only if H is obtained from a hypertree by doubling its edges.This result reminiscent of Berge's and Lovász's si...

A graph $G$ of order $n$ is pancyclic if it contains cycles of all lengths from 3 to $n$. A graph is called cycle extendable if for every cycle $C$ of less than $n$ vertices there is another cycle $C^*$ containing all vertices of $C$ plus a single new vertex. Clearly, every cycle extendable graph is pancyclic if it contains a triangle. Cycle extend...

In this note, we consider a minimum degree condition for a hamiltonian graph to have a 2-factor with two components. Let G be a graph of order n⩾3. Dirac's theorem says that if the minimum degree of G is at least , then G has a hamiltonian cycle. Furthermore, Brandt et al. [J. Graph Theory 24 (1997) 165–173] proved that if n⩾8, then G has a 2-facto...

Given positive integers k m n, a graph G of order n is (k, m)-pancyclic ordered if for any set of k vertices of G and any integer r with m r n, there is a cycle of length r encountering the k vertices in a specified order. Minimum degree conditions that imply a graph of sufficiently large order n is (k, m)-pancylic ordered are proved. Examples show...

In [Discrete Math. 251, No. 1–3, 71–76 (2002; Zbl 1002.05044)], J. Brousek characterizes all triples of connected graphs, G 1 ,G 2 ,G 3 , with G i =K 1,3 for some i=1 ,2, or 3, such that all G 1 G 2 G 3 -free graphs contain a hamiltonian cycle. In [Discrete Math. 249, No. 1–3, 71–81 (2002; Zbl 0990.05091)], R. J. Faudree, R. J. Gould, M. S. G. Jaco...

In this paper we generalize a Theorem of Jung which shows that 1-tough graphs with δ(G)⩾|V(G)|-42 are hamiltonian. Our generalization shows that these graphs contain a wide variety of 2-factors. In fact, these graphs contain not only 2-factors having just one cycle (the hamiltonian case) but 2-factors with k cycles, for any k such that 1⩽k⩽n-164.

Given positive integers kmn, a graph G of order n is (k,m)-pancyclic if for any set of k vertices of G and any integer r with mrn, there is a cycle of length r containing the k vertices. Minimum degree conditions and minimum sum of degree conditions of nonadjacent vertices that imply a graph is (k,m)-pancylic are proved. If the additional property...

The bar visibility number of a graph G, denoted b(G), is the minimum t such that G can be represented by assigning each vertex x the set S x of points in at most t horizontal segments in the plane so that uv∈E(G) if and only if some point of S u sees some point of S v via a vertical segment of positive width unobstructed by assigned points. Among o...

For a graph G, a positive integer k, k ≥ 2, and a non-negativeinteger with z < k and z / = 1, a subset D of the vertex set V (G) is said to be a non-z (mod k) dominating set if D is a dominating set and for all x ∈ V (G), |N [x] ∩ D| /≡ z (mod k).For the case k = 2 and z = 0, it has been shown that these sets exist for all graphs. The problem for k...

A graph is called fragile if it has a vertex cut which is also an independent set. Chen and Yu proved that every graph with n vertices and at most 2n−4 edges is fragile, which was conjectured to be true by Caro. However, their proof does not give any information on the number of vertices in the independent cuts. The purpose of this paper is to inve...

In this paper we characterize those forbidden triples of graphs, no one of which is a generalized claw, sufficient to imply that a 2-connected graph of sufficiently large order is hamiltonian.

In this paper we characterize those forbidden triples of graphs, no one of which is a generalized claw, sufficient to imply that a 2-connected graph of sufficiently large order is hamiltonian.

Let N(n,k) be the minimum number of pairwise edge disjoint monochromatic complete graphs Kk in any 2-coloring of the edges of a Kn. Upper and lower bounds on N(n,k) will be given for k⩾3. For k=3, exact values will be given for n⩽11, and these will be used to give a lower bound for N(n,3).

Let G be a graph of order n. Define fk(G) (Fk(G)) to be the minimum (maximum) number of components in a k-factor of G. For convenience, we will say that fk(G)=0 if G does not contain a k-factor. It is known that if G is a claw-free graph with sufficiently high minimum degree and proper order parity, then G contains a k-factor. In this paper we show...

We construct all six-element orders which are not 50%-tolerance orders. We show that a width-two order is a 50% tolerance order if and only if no restriction of the order to a six-element set is isomorphic to one of these six-element orders. This yields a corresponding characterization of bipartite 50%-tolerance graphs. Since an order (graph) has a...

Let N(n,k) be the minimum number of pairwise edge disjoint monochromatic complete graphs Kk in any 2-coloring of the edges of a Kn. Upper and lower bounds on N(n,k) will be given for k⩾3. For k=3, exact values will be given for n⩽11, and these will be used to give a lower bound for N(n,3).

A graph G is a (P 1 ,P 2 )-intersection graph of a graph if there exists a host graph H and a family of 1-paths and 2-paths in H such that G is the intersection graph of these paths. The (P 1 ,P 2 )-intersection graphs of a tree and a path are characterized by forbidden subgraphs. Also the (P 1 ,P 2 ,P 3 )-intersection graphs of a path are characte...

In a graph, a set D is an n-dominating set if for every vertex x, not in D, x is adjacent to at least n vertices of D. The n-domination number, γ n (G), is the order of a smallest n-dominating set. When this concept was first introduced by Fink and Jacobson, they asked whether there existed a function f(n), such that if G is any graph with minimum...

It will be shown that if G is a graph of order n which contains a triangle, a cycle of length n or n−1 and at least cn odd cycles of different lengths for some positive constant c, then there exists some positive constant k=k(c) such that G contains at least kn 1/6 even cycles of different lengths. Other results on the number of even cycle lengths...

Ng and Schultz [J Graph Theory 1 (1997), 45–57] introduced the idea of cycle orderability. For a positive integer k, a graph G is k-ordered if for every ordered sequence of k vertices, there is a cycle that encounters the vertices of the sequence in the given order. If the cycle is also a Hamiltonian cycle, then G is said to be k-ordered Hamiltonia...

Ng and Schultz [J Graph Theory 1 (1997), 45–57] introduced the idea of cycle orderability. For a positive integer k, a graph G is k-ordered if for every ordered sequence of k vertices, there is a cycle that encounters the vertices of the sequence in the given order. If the cycle is also a Hamiltonian cycle, then G is said to be k-ordered Hamiltonia...

We prove that a locally cobipartite graph on n vertices contains a family of at most n cliques that cover its edges. This is related to Opsut's conjecture that states the competition number of a locally cobipartite graph is at most two.

. In the study of hamiltonian graphs, many well known results use degree conditions to ensure sufficient edge density for the existence of a hamiltonian cycle. Recently it was shown that the classic degree conditions of Dirac and Ore actually imply far more than the existence of a hamiltonian cycle in a graph G, but also the existence of a 2-factor...

The girth of a graph with a Hamiltonian cycle and t chords is investigated. In particular, for any integer t>0 let g(t) denote the smallest number such that any Hamiltonian graph G with n vertices and n+t edges has girth at most g(t)n+c, where c is a constant independent of n. It is shown that there exist constants c 1 and c 2 such that (c 1 (logt)...

Moon and Moser (Israel J. Math. 1 (1962) 163-165) showed that if G is a balanced bipartite graph of order 2n and minimum degree delta greater than or equal to (n + 1)/2, then G is hamiltonian. Recently, it was shown that their well-known degree condition also implies the existence of a 2-factor with exactly k cycles provided n greater than or equal...

A path of a graph is maximal if it is not a proper subpath of any other path of the graph. A graph is scenic if every maximal path of the graph is a maximum length path. In [Scenic graphs. I: Traceable graphs. Ars Comb. 49, 79-96 (1998)], we give a new proof of C. Thomassen’s result characterizing all scenic graphs with Hamiltonian path. (Compare C...

We translate Ramsey-type problems into the language of decomposable hereditary properties of graphs. We prove a distributive law for reducible and decomposable properties of graphs. Using it we establish some values of graph theoretical invariants of decomposable properties and show their correspondence to generalized Ramsey numbers.

In this note we prove that every 2-connected graph of order n with no repeated cycle lengths has at most edges and we show this result is best possible with the correct order of magnitude on $\sqrt{n}$. The 2-connected case is also used to give a quick proof of Lai's result on the general case. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 11–...

. We introduce the visibility number b(G) of a graph G, which is the minimum t such that G can be represented by assigning each vertex a union of at most t horizontal segments in the plane so that vertices u; v are adjacent if and only if some point assigned to u sees some point assigned to v via a vertical segment unobstructed by other assigned po...

A path of a graph is maximal if it is not a proper subpath of any other path of the graph. The path spectrum is the set of lengths of all maximal paths in the graph. A graph is scenic if its path spectrum is a singleton set. In this paper we give a new proof characterizing all scenic graphs with a Hamiltonian path; this result was first proven by T...