A graph is called "triangle-free" if it does not contain a cycle of length three (a cycle with three edges). Prove that any triangle-free planar graph can be coloured by 4 colours.
Can someone show me how to prove this problem?
2026-03-29 18:33:06.1774809186
Prove that any triangle-free planar graph can be colored by 4 colors.
134 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail AtRelated Questions in GRAPH-THEORY
- characterisation of $2$-connected graphs with no even cycles
- Explanation for the static degree sort algorithm of Deo et al.
- A certain partition of 28
- decomposing a graph in connected components
- Is it true that if a graph is bipartite iff it is class 1 (edge-coloring)?
- Fake induction, can't find flaw, every graph with zero edges is connected
- Triangle-free graph where every pair of nonadjacent vertices has exactly two common neighbors
- Inequality on degrees implies perfect matching
- Proving that no two teams in a tournament win same number of games
- Proving that we can divide a graph to two graphs which induced subgraph is connected on vertices of each one
Related Questions in COLORING
- Name of Theorem for Coloring of $\{1, \dots, n\}$
- Is it true that if a graph is bipartite iff it is class 1 (edge-coloring)?
- Orbit counting lemma hexagon
- difference between colouring number and chromatic number
- Is it a tetrahedron, 5-cell, or something else?
- Distance of closest neighbor points in a vectorspace ${\mathbb R}^n$ (infinitesimal or zero)?
- How to uniquely label a connected graph?
- Graph coloring: $G$ is a graph where the number of vertices with degree of at least $k$, is at most $k$. Prove $χ(G) \le k$
- Complete graphs in the plane with colored edges where an edge don't cross edges with same color
- 4-chromatic unit distance graph with no 4-cycles.
Related Questions in PLANAR-GRAPHS
- max planarity and triangulation
- How to find a minimal planar covering of a graph
- Complete bipartite planar graphs
- Is graph Planar?
- Kuratowski's theorem on Planar graphs
- Graph Theory: Degree of regions in a planar graph
- On the Hex/Nash connection game theorem
- Graph Theory: There is only one planar graph with all vertices of degree 4 and 10 regions
- Intersection of circles and regions
- Which of these graphs is planar?
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?