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198 results on '"JACOBSON, MICHAEL S."'

2. Graph Saturation in Multipartite Graphs

Author

Ferrara, Michael, Jacobson, Michael S., Pfender, Florian, and Wenger, Paul S.

Subjects
Mathematics - Combinatorics
Abstract

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 maximum and minimum size of an ${\mathcal F}$-saturated subgraph of $G$, respectively. If no element of ${\mathcal F}$ is a subgraph of $G$, then $\text{sat}({\mathcal F},G) = \text{ex}({\mathcal F}, G) = |E(G)|$. In this paper, for $k\ge 3$ and $n\ge 100$ we determine $\text{sat}(K_3,K_k^n)$, where $K_k^n$ is the complete balanced $k$-partite graph with partite sets of size $n$. We also give several families of constructions of $K_t$-saturated subgraphs of $K_k^n$ for $t\ge 4$. Our results and constructions provide an informative contrast to recent results on the edge-density version of $\text{ex}(K_t,K_k^n)$ from [A. Bondy, J. Shen, S. Thomass\'e, and C. Thomassen, Density conditions for triangles in multipartite graphs, Combinatorica 26 (2006), 121--131] and [F. Pfender, Complete subgraphs in multipartite graphs, Combinatorica 32 (2012), no. 4, 483--495]., Comment: 16 pages, 4 figures

Published
2014

3. Gaps in the Saturation Spectrum of Trees

Author

Horn Paul, Gould Ronald J., Jacobson Michael S., and Thomas Brent J.

Subjects
saturation spectrum, tree, saturation number, 05c35, 05c05, Mathematics, QA1-939
Abstract

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 set of sizes of H-saturated graphs between sat(n,H) and ex(n,H). In this paper we show that paths, trees with a vertex adjacent to many leaves, and brooms have a gap in the saturation spectrum.

Published
2019
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4. Saturation Spectrum of Paths and Stars

Author

Faudree Jill, Faudree Ralph J., Gould Ronald J., Jacobson Michael S., and Thomas Brent J.

Subjects
saturation spectrum, stars, paths, 05c35, 05c05, Mathematics, QA1-939
Abstract

A graph G is H-saturated if H is not a subgraph of G but the addition of any edge from G̅ 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 sat(n,H) and ex(n,H). In this paper we completely determine the saturation spectrum of stars and we show the saturation spectrum of paths is continuous from sat(n, Pk) to within a constant of ex(n, Pk) when n is sufficiently large.

Published
2017
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7. To Infinity. . .and beyond!

Author

Larson, Jeffrey M., Jacobson, Michael S., and Den Ouden, Katherine M.

Abstract

Developmentally, middle school students progress from being concrete thinkers and learners to abstract thinkers. Traditional middle school mathematics instruction introduces a curriculum that is intuitive and taught within a natural trajectory of the content. However, with this traditional approach, students may miss out on being exposed to challenging and complex topics from advanced mathematics that can greatly expand the way that students think mathematically. Concepts such as infinite sets can be counterintuitive, but they can hold the interest of most students. Perhaps this is because of their inherent eccentricity, which is counter to students' natural ability to easily adapt to finite objects and events. Although there are various avenues to introduce these complex mathematical topics, the authors propose an activity about the concept of infinity that highlights the importance of proof and reasoning and expands the learner's abstract thinking. (Contains 1 figure.)

Published
2012

9. Making Meaning through Mentorship: A Student-Led Layered Peer Mentorship Program.

Author

Cummings, Miriam Howland, Schupbach, William Taylor, Altman, Tom, Jacobson, Michael S., Goodman, Katherine, and Darbeheshti, Maryam

Abstract

This Complete Evidence-Based Paper presents research about a layered peer mentorship program for undergraduate engineering students at a public urban research university and ways that students have made meaning from their mentorship experiences. This mentorship program began in Fall 2019 and has grown to include the following layers: (a) first-year students who receive mentorship, (b) sophomore- and junior-level students who serve as mentors (all of whom received mentorship during their first year), (c) junior- and senior-level students who serve as lead mentors who design the program for that academic year (including content, group meetings, service projects, meeting schedules, etc.), (d) a graduate student who mentors and supervises the lead mentors, and (e) a faculty member who oversees the overall program, provides general guidance, and advises all the students. We will describe ways in which the participating students have made meaning of their experience in the program, highlighting three key areas: (1) the web of relationships formed, which cohere into a community; (2) students' transitions from receiving mentorship as first-year students to mentoring others in their sophomore and junior years; and (3) the feedback and iteration process by which the program has continuously developed, which forefronts student voice and agency. The paper will provide specific examples in each of the three key areas described, with a special focus on students' own descriptions of the meaning they have made through their participation in the mentorship program. Recommendations will also be shared for those interested in implementing similar programs on their campuses. The mentorship program forms a complex web of relationships between and among students, and these relationships cohere into a felt community for students. These relationships are forged by various means in various contexts. First, each student who receives mentorship is a first-year student in the Engineering Learning Community (ELC). In the ELC, first-year students take a bundle of courses together, with specific sections of English composition, math, and a first-year design innovations course. Second, the students who receive mentorship are assigned peer mentors who are sophomore- and junior-level students who were previously in the ELC. Mentees and mentors meet regularly in both individual and group meetings, which affords both individual and group relationships to form. All these interplaying individual and group relationships form a rich, complex web which coheres into a community for students. Additionally, students also find meaning through their transition from mentee in their first year to mentor in their sophomore and junior years. This meaning is forged through personal growth, as well as through a reification of engineering identity when they are viewed as legitimate members of the community of engineering majors. When students become mentors, they transition into a position where first-year students look to them for advice, support, and encouragement. Mentors enact their identity as someone with experience who has valuable perspective to share. Mentors can feel more confident in their classes knowing that their status as an engineering major is respected and has been legitimated. Lastly, serving as a mentor is something students look forward to listing on their resume as a demonstration of teamwork and people skills which engineering employers so highly value. A third way students make meaning from the mentorship program is through the feedback and iteration process by which the program is continuously formed. At the end of each semester, students submit formal, anonymous, formative feedback on the program. The suggestions students make are implemented almost immediately, making this mentorship program responsive and adaptable. Students in the program know that their feedback is taken seriously, and that they have the power to create the mentorship program they find most helpful. Additionally, the lead mentors have nearly complete autonomy to organize, schedule, and run the program as they see fit. The upper layers of the program - the graduate student and the faculty member - provide support, advice, and resources, but the undergraduate students are the ones iterating and forming the program. Through these means, the program has evolved in several ways, including the introduction of group meetings and a semesterly service project, which were the direct implementations of student ideas. This focus on student voice and agency enables students to build self-efficacy and make meaning from their experiences with the mentorship program. [ABSTRACT FROM AUTHOR]

Published
2023

10. Weak Saturation Numbers for Sparse Graphs

Author

Faudree Ralph J., Gould Ronald J., and Jacobson Michael S.

Subjects
saturated graphs, sparse graphs, weak saturation, Mathematics, QA1-939
Abstract

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 time, each edge added creates a new copy of F. The minimum size of a weakly F-saturated graph G of order n will be denoted by wsat(n, F). The graphs of order n that are weakly F-saturated will be denoted by wSAT(n, F), and those graphs in wSAT(n, F) with wsat(n, F) edges will be denoted by wSAT(n, F). The precise value of wsat(n, T) for many families of sparse graphs, and in particular for many trees, will be determined. More specifically, families of trees for which wsat(n, T) = |T|−2 will be determined. The maximum and minimum values of wsat(n, T) for the class of all trees will be given. Some properties of wsat(n, T) and wSAT(n, T) for trees will be discussed. Keywords: saturated graphs, sparse graphs, weak saturation.

Published
2013
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14. Three Years of the Urban STEM Collaboratory.

Author

Darbeheshti, Maryam, Cummings, Miriam Howland, Ivey, Stephanie S., Russomanno, David J., Alfrey, Karen D., Watt, Jeffrey, Stewart, Craig O., Jacobson, Michael S., Altman, Tom, and Goodman, Katherine

Subjects
STEM education, URBAN research, SOCIAL networks
Published
2022

34. Degree sum and vertex dominating paths.

Author

Faudree, Jill, Faudree, Ralph J., Gould, Ronald J., Horn, Paul, and Jacobson, Michael S.

Subjects
PATHS & cycles in graph theory, MATHEMATICAL proofs, GEOMETRIC vertices, SET theory, LOGARITHMIC functions
Abstract

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 σ(k+2)(G)≥n−2k−1 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 such that σ2(G)≥2nk+2+f(k) contains a vertex dominating path of length at most (20k)|T|, where T is a minimum dominating set of vertices. An immediate corollary of this result is that every such graph contains a vertex dominating path with length bounded above by a logarithmic function of the order of the graph. To derive this result, we prove that every n‐vertex k‐connected graph with σ2(G)≥2nk+2+f(k) contains a path of length at most 20k|T|, through any set of T vertices where |T|≤n/900k4. [ABSTRACT FROM AUTHOR]

Published
2018
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35. Minimum Degree and Dominating Paths.

Author

Faudree, Ralph J., Gould, Ronald J., Jacobson, Michael S., and West, Douglas B.

Subjects
GEOMETRIC vertices, LOGARITHMIC curves, LOGARITHMS, GRAPH theory, SPANNING trees
Abstract

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 [ABSTRACT FROM AUTHOR]

Published
2017
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40. Packing of graphic n‐tuples

Author

Busch, Arthur H., primary, Ferrara, Michael J., additional, Hartke, Stephen G., additional, Jacobson, Michael S., additional, Kaul, Hemanshu, additional, and West, Douglas B., additional

Published
2011
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42. Chvátal-Erdös type theorems

Author

Faudree, Jill R., primary, Faudree, Ralph J., additional, Gould, Ronald J., additional, Jacobson, Michael S., additional, and Magnant, Colton, additional

Published
2010
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