A graph is called color-critical provided each subgraph...

A graph is called color-critical provided each subgraph obtained by removing a vertex has a smaller chromatic number. Let $G=(V, E)$ be a color-critical graph. Prove the following: (a) $\chi\left(G_{V-\{x\}}\right)=\chi(G)-1$ for every vertex $x$. (b) $G$ is connected. (c) Each vertex of $G$ has degree at least equal to $\chi(G)-1$. (d) $G$ does not have an articulation set $U$ such that $G_{U}$ is a complete graph. (e) Every graph $H$ has an induced subgraph $G$ such that $\chi(G)=\chi(H)$ and $G$ is color-critical.

A graph is called color-critical provided each subgraph obtained by removing a vertex has a smaller chromatic number. Let $G=(V, E)$ be a color-critical graph. Prove the following:
(a) $\chi\left(G_{V-\{x\}}\right)=\chi(G)-1$ for every vertex $x$.
(b) $G$ is connected.
(c) Each vertex of $G$ has degree at least equal to $\chi(G)-1$.
(d) $G$ does not have an articulation set $U$ such that $G_{U}$ is a complete graph.
(e) Every graph $H$ has an induced subgraph $G$ such that $\chi(G)=\chi(H)$ and $G$ is color-critical.

Key Concepts

Induced Subgraph

An induced subgraph is formed by choosing a subset of vertices from the original graph and including every edge that connects pairs of vertices within that subset. Studying induced subgraphs is important because they inherit many of the parent graph’s properties, such as the chromatic number, and are used to identify core structures within larger graphs.

Complete Graph

A complete graph is one in which every pair of distinct vertices is connected by an edge, making it the maximally connected graph on a given set of vertices. The complete graph has a chromatic number equal to its number of vertices, and it serves as an extreme case in graph coloring problems and in the study of color-critical graphs due to its very rigid structure.

Articulation Set

An articulation set in a graph is a set of vertices whose removal increases the number of connected components of the graph. These vertices are critical points whose absence fragments the graph. In problems involving color-critical graphs, restrictions on articulation sets—especially when they form a complete subgraph—help in ensuring that the graph remains minimally structured to enforce its chromatic properties.

Graph Connectivity

Graph connectivity refers to the property that there is a path between any two vertices in the graph. A connected graph does not break into isolated subgraphs, ensuring the integrity of the overall structure. This concept is critical when analyzing color-critical graphs, as connectivity guarantees that the effect of removing any vertex impacts the entire graph.

Vertex Degree

The degree of a vertex is the number of edges incident to it, indicating how many direct connections it has with other vertices. In many graph-theoretical contexts, including color-critical graphs, a lower bound on vertex degree (often related to the chromatic number) ensures that each vertex significantly contributes to the overall structure of the graph.

Chromatic Number

The chromatic number of a graph is the minimum number of distinct colors required to color the vertices so that no two adjacent vertices share the same color. This concept is fundamental in graph theory, as it measures the complexity of a graph’s structure in terms of vertex adjacency and is used to study coloring problems and graph classification.

Color-Critical Graph

A color-critical graph is a graph in which the removal of any vertex decreases its chromatic number. This means that every vertex is crucial in maintaining the graph's chromatic value, and each vertex's presence is necessary for the graph to require that many colors in any proper vertex coloring. It is a key concept in understanding minimal structures that enforce high chromatic numbers.

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