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Graphs of functions provide a visual representation of how output values change in response to varying inputs. Each point on the graph corresponds to an ordered pair, where the x-coordinate (independent variable) determines the horizontal position and the y-coordinate (dependent variable) determines the vertical position. Linear functions like y = x give a straight line, indicating a constant rate of change.Nonlinear functions display more complex behaviors. Even power functions generate...
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Trigonometric functions exhibit periodic and symmetrical behavior, deeply rooted in the unit circle. The sine and cosine functions correspond to the vertical and horizontal projections, respectively, of a point rotating counterclockwise around the circle. These functions trace smooth, repeating waveforms with identical periods and bounded ranges. The tangent function is defined as the ratio of sine to cosine and produces an unbounded curve that repeats every units, with vertical asymptotes...
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Subdivision of graphs in .

Kristiana Wijaya1, Edy Tri Baskoro2, Hilda Assiyatun2

  • 1Graph, Combinatorics, and Algebra Research Group, Department of Mathematics, FMIPA, Universitas Jember, Jalan Kalimantan 37 Jember 68121, Indonesia.

Heliyon
|July 9, 2020
PubMed
Summary

Researchers introduce a novel method to construct new Ramsey minimal graphs. This subdivision operation on existing graphs generates previously unknown Ramsey minimal graph structures, expanding the field of Ramsey theory.

Keywords:
MatchingMathematicsPathRamsey minimal graphsRed-blue coloring

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Area of Science:

  • Graph Theory
  • Combinatorics
  • Ramsey Theory

Background:

  • Ramsey theory studies the conditions under which order must appear in a large structure.
  • Ramsey-minimal graphs are fundamental in understanding the boundary cases of Ramsey theorems.
  • The notation G ↠ (H, K) signifies that any edge coloring of G contains a monochromatic copy of H or K.

Purpose of the Study:

  • To develop a new construction method for generating Ramsey minimal graphs.
  • To expand the known set of Ramsey minimal graphs using a subdivision operation.
  • To investigate the properties of Ramsey minimal graphs derived from existing ones.

Main Methods:

  • A novel subdivision operation is proposed for constructing new Ramsey minimal graphs.
  • The construction involves subdividing a specific edge within a cycle of a known Ramsey minimal graph.
  • The method focuses on creating a new Ramsey minimal graph from a given graph F by subdividing an edge e four times.

Main Results:

  • A new, simple construction for generating Ramsey minimal graphs is presented.
  • The subdivision operation successfully creates new Ramsey minimal graphs from existing ones.
  • The paper demonstrates the creation of a new Ramsey minimal graph by subdividing an edge in a cycle four times.

Conclusions:

  • The proposed subdivision operation is an effective method for constructing novel Ramsey minimal graphs.
  • This research contributes to the understanding and cataloging of Ramsey minimal graphs.
  • The findings open avenues for further exploration of graph structures within Ramsey theory.