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Chapter 11: Problem 19

Let \(G=(V, E)\) be a loop-free undirected graph. We call \(G\) color-critical if \(\chi(G)>\chi(G-v)\) for all \(v \in V\). a) Explain why cycles with an odd number of vertices are color-critical while cycles with an even number of vertices are not color-critical. b) For \(n \in \mathbf{Z}^{+}, n \geq 2\), which of the complete graph \(K_{n}\) are color-critical? c) Prove that a color-critical graph must be connected. d) Prove that if \(G\) is color-critical with \(\chi(G)=k\), then \(\operatorname{deg}(v) \geq k-1\) for all \(v \in V\)

Short Answer

Expert verified
a) Odd cycles are color-critical as removing a vertex decreases their chromatic number. Even cycles are not color-critical as removing a vertex doesn't change their chromatic number. b) Complete graphs \( K_{n} \) are color-critical when \( n \geq 3 \). c) A color-critical graph must be connected. This can be proven by contradiction assuming that a color-critical graph is disconnected. d) If \( G \) is color-critical with chromatic number \( k \), every vertex \( v \) in \( G \) must have degree at least \( k - 1 \). This can be proven by contradiction assuming there exists a vertex \( v \) with \( \operatorname{deg}(v) < k-1 \).

Step by step solution

01

Explain Color-Criticality in Cycles

a) A graph \( G \) is color-critical if the removal of any vertex decreases its chromatic number. An odd cycle, say \( C_{2n+1} \), contains \(2n + 1\) vertices. Due to this odd number of vertices, we have to use at least 3 colors so that it meets the Graph Coloring rules. However, once a single vertex is removed, it becomes an even cycle, which can be colored with only two colors. Hence, odd cycles are color-critical. In contrast, for an even cycle, say \( C_{2n} \), it can be colored with only two colors. Even if we remove a vertex, the resultant graph still requires two colors. So it's not color-critical.
02

Determine Color-Criticality in Complete Graphs

b) The complete graph \( K_{n} \) is color-critical only when \( n \geq 3 \). This is because complete graphs with two vertices can be colored with two colors, and removing a vertex makes them colorable by one color. Meanwhile, any complete graph with \( n \geq 3 \) vertices will require \( n \) colors, and removing any vertex results in \( K_{n-1} \), which is colorable with \( n-1 \) colors. Hence, \( K_{n} \) with \( n \geq 3 \) are color-critical.
03

Prove Connectivity in Color-Critical Graphs

c) To prove that a color-critical graph must be connected, we'll assume that a color-critical graph is disconnected. That means it is composed of two disjoint sub-graphs, each with its own chromatic number. The chromatic number of the entire graph would be the maximum of the two sub-graphs. If we remove a vertex from the sub-graph with smaller chromatic number, the chromatic number of the entire graph doesn't change, contradicting the definition of color-critical graphs. Thus, a color-critical graph must be connected.
04

Prove Degree Property in Color-Critical Graphs

d) If \( G \) is color-critical with chromatic number \( k \), then every vertex \( v \) in \( G \) must have degree at least \( k - 1 \). Assume to the contrary that a graph \( G \) is color-critical with \( \chi(G) = k \), and there exists a vertex \( v \) with \( \operatorname{deg}(v) < k-1 \). Then, \( G-v \) needs only \( k-1 \) colors (since vertex \( v \) had at least one color not adjacent to it in the graph \( G \)), which contradicts with fact that \( \chi(G) > \chi(G-v) \). Thus, if \( \chi(G) = k \), every vertex in \( G \) must have a degree of at least \( k-1 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Color-Critical Graphs
Color-critical graphs are those fascinating structures in the field of graph theory that have a special property. If you remove any vertex from a color-critical graph, its chromatic number decreases. This makes each vertex crucial for maintaining its chromatic color count.

  • Odd cycles, such as a cycle with 5 vertices, exemplify this trait because they switch from needing 3 colors to only 2 once a vertex is removed, turning the cycle into an even length.
  • Contrarily, even cycles always require only 2 colors. Thus, removing a vertex does not reduce the chromatic number, keeping them from being color-critical.
Overall, color-critical graphs are instrumental for understanding how graph topology influences their chromatic properties.
Chromatic Number
The chromatic number of a graph is a key metric in graph coloring. It represents the smallest number of colors needed to color a graph such that no two adjacent vertices share the same color.

  • For instance, complete graphs with at least 3 vertices have chromatic numbers equal to their vertex count, meaning a complete graph with 5 vertices will require 5 colors.
  • Understanding chromatic numbers helps in scenarios like scheduling and timetable designs where conflict-free assignments are crucial.
Exploring chromatic numbers in various graph structures allows us to grasp how constraints on vertex connections demand more diverse color allocations.
Complete Graphs
Complete graphs are a special kind of graph where every pair of distinct vertices are connected by a unique edge. This characteristic makes them particularly demanding in terms of chromatic numbers.

  • Complete graphs, denoted as \(K_n\), require exactly \(n\) different colors for \(n\) vertices, because each vertex is adjacent to every other vertex, necessitating a unique color for each.
  • These graphs become color-critical when they have 3 or more vertices, as removing a vertex reduces their chromatic number by one.
The study of complete graphs is essential, as their fully connected nature pushes the limits of graph coloring principles.
Connected Graphs
A graph is connected if there is a path between every pair of vertices, making connected graphs crucial in maintaining graph unity.

  • In the context of color-critical graphs, connectivity ensures that removing a vertex affects the entire graph. If a graph were disconnected, removing a vertex from a disconnected part wouldn鈥檛 change its overall chromatic number.
  • This concept reinforces the notion that for a graph to be color-critical, it must also be connected, highlighting the interdependence of graph structure and coloring properties.
Understanding connected graphs sheds light on how the relationships within a graph influence its behavior in graph coloring tasks.

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Most popular questions from this chapter

a) Suppose that the complete bipartite graph \(K_{m, n}\) contains 16 edges and satisfies \(m \leq n\). Determine \(m, n\) so that \(K_{m, n}\) possesses (i) an Euler circuit but not a Hamilton cycle; (ii) both a Hamilton cycle and an Euler circuit. b) Generalize the results of part (a).

Let \(G=(V, E)\) be a directed graph, where \(|V|=n\) and \(|E|=e\). What are the values for \(\sum_{v \in V} i d(v)\) and \(\sum_{v \in V} \operatorname{od}(v) ?\)

a) Find a graph \(G\) where both \(G\) and \(\bar{G}\) are connected. b) If \(G\) is a graph on \(n\) vertices, for \(n \geq 2\), and \(G\) is not connected, prove that \(\bar{G}\) is connected.

Determine \(|V|\) for the following graphs or multigraphs \(G\). a) \(G\) has nine edges and all vertices have degree 3 . b) \(G\) is regular with 15 edges. c) \(G\) has 10 edges with two vertices of degree 4 and all others of degree 3 .

If \(G=(V, E)\) is a connected graph with \(|E|=17\) and \(\operatorname{deg}(v) \geq 3\) for all \(v \in V\), what is the maximum value for \(|V| ?\)
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