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Graphical Representation of Inequalities01:28

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The graph of the equation where y equals x squared forms a curve known as a parabola. This curve acts as a boundary in the coordinate plane, dividing it into distinct regions based on the relative position of points.When the equality sign in the equation is replaced with an inequality—such as greater than, less than, greater than or equal to, or less than or equal to—the graphical representation changes from a single curve into a broader shaded area that signifies the set of all...
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Vectors can be multiplied by scalars, added to other vectors, or subtracted from other vectors. The vector sum of two (or more) vectors is called the resultant vector or, for short, the resultant.
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The polar coordinate system represents points using a distance from a central point (the pole) and an angle from a reference direction (the polar axis). Unlike rectangular coordinates, polar coordinates are ideal for graphing curves with radial symmetry or periodic behavior.Some general forms of graphs in polar coordinates include the following:Equation of a Circle (Centered at the Pole):A graph where the radius remains constant for all angles traces a circle centered at the pole:Equation of a...
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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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An equation with two variables, typically written in the form y = f(x) or Ax + By = C, describes a relationship between quantities represented by x and y. Each solution to such an equation is an ordered pair (x, y) that satisfies the equation when substituted. These pairs can be represented graphically to understand the variables' relationship visually.A common technique for constructing the graph of a two-variable equation is to create a value table. Begin by choosing several values for the...
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Solving Inequalities Graphically01:24

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Solving inequalities graphically involves using a visual approach to determine where a mathematical expression meets a specific condition, such as being greater than or less than another value. By examining the position of a graph relative to the x-axis or another graph, it becomes possible to identify the range of x-values that satisfy the inequality. This method provides an intuitive understanding of solution intervals by showing where the inequality holds true.Graphical solutions to...
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Related Experiment Video

Updated: Dec 7, 2025

Generating Strictly Controlled Stimuli for Figure Recognition Experiments
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Coloring triangle-free graphs with local list sizes.

Ewan Davies1, Rémi de Joannis de Verclos2, Ross J Kang2

  • 1Korteweg-De Vries Institute for Mathematics University of Amsterdam Amsterdam Netherlands.

Random Structures & Algorithms
|October 1, 2020
PubMed
Summary
This summary is machine-generated.

This study refines results on the list chromatic number of triangle-free graphs by allowing lower color bounds for low-degree vertices. Findings impact list, correspondence, and fractional coloring theories.

Keywords:
hard‐core modellist coloringtriangle‐free graphs

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Area of Science:

  • Graph Theory
  • Combinatorics
  • Discrete Mathematics

Background:

  • Recent breakthroughs by Molloy and Bernshteyn advanced understanding of the list chromatic number in triangle-free graphs.
  • The list chromatic number is a key concept in graph coloring, measuring the minimum number of colors needed for a proper coloring from given lists of colors at each vertex.

Purpose of the Study:

  • To provide two distinct and natural refinements of existing results on the list chromatic number of triangle-free graphs.
  • To explore relaxations in color availability for vertices with lower degrees.
  • To investigate implications for list coloring, correspondence coloring, and fractional coloring.

Main Methods:

  • Development of two novel refinement techniques for list coloring bounds.
  • Application of the hard-core model to prove a Johansson-type result in fractional coloring.
  • Analysis of color availability adjustments based on vertex degree.

Main Results:

  • Two new refinements of the list chromatic number for triangle-free graphs are established.
  • A result concerning list coloring and correspondence coloring is presented.
  • A new result in fractional coloring is demonstrated, utilizing the hard-core model.

Conclusions:

  • The study offers more nuanced bounds for coloring triangle-free graphs, particularly for vertices with lower degrees.
  • The techniques developed, especially the use of the hard-core model, may have broader applications in graph theory and related fields.