Star coloring in graph theory is a specialized type of graph coloring that adheres to stricter rules than a simple proper coloring. It is defined as a proper coloring of a graph such that no path of length 3 in the graph is bicolored. This rigorous condition ensures a more dispersed and intricate distribution of colors across the graph's vertices.
Key Conditions for a Star Coloring
A coloring of a graph $G$ is considered a star coloring if it satisfies two essential conditions:
- Proper Coloring: As with any standard graph coloring, adjacent vertices (vertices connected by an edge) must be assigned different colors. This prevents ambiguity and ensures clear differentiation between directly connected elements.
- No Bicolored Path of Length 3: This is the defining and more restrictive condition.
- A "path of length 3" refers to a sequence of four distinct vertices and three edges connecting them, e.g., $v_1 - v_2 - v_3 - v_4$.
- "Bicolored" means that only two colors are used for the vertices in this path, and they alternate. For instance, if colors are A and B, a bicolored path of length 3 would look like A-B-A-B or B-A-B-A.
- Therefore, a star coloring prohibits any such alternating two-color pattern on any path of four vertices in the graph. This powerful condition implicitly ensures that no two vertices at distance 2 can have the same color. For any path $u-v-w$, $color(u)$ must be different from $color(w)$. If $color(u)$ and $color(w)$ were the same (e.g., $C_1-C_2-C_1$), and this path were extended to a $P_4$ ($C_1-C_2-C_1-C_2$), it would be a forbidden bicolored path of length 3.
The Star Chromatic Number ($\chi_s(G)$)
The star chromatic number, denoted as $\chi_s(G)$, represents the minimum number of colors required to achieve a star coloring of a graph $G$. It is a fundamental parameter in the study of star colorings, analogous to the chromatic number ($\chi(G)$) for proper colorings.
Why "Star"?
The name "star coloring" originates from the structure it prohibits. A path of length 2 ($P_3$), consisting of three vertices $u-v-w$, can be thought of as a "star" subgraph where $v$ is the center and $u, w$ are its leaves. The condition that $color(u) \neq color(w)$ means that the leaves of any such "star" must always have different colors.
Star Coloring vs. Other Graph Colorings
Star coloring is a specific type of graph coloring that is generally more restrictive than other forms:
| Coloring Type | Description | Key Condition(s) | Relationship |
|---|---|---|---|
| Proper Coloring | Assigns colors to vertices such that no two adjacent vertices share the same color. | Adjacent vertices have different colors. | Most basic; all other types are also proper colorings. |
| Star Coloring | A proper coloring where no path of length 3 is bicolored (implying no two vertices at distance 2 share a color). | Adjacent vertices different; No $P_4$ of type A-B-A-B; endpoints of any $P_3$ different. | Stronger than proper coloring and acyclic coloring. |
| Acyclic Coloring | A proper coloring where every cycle contains at least three distinct colors (i.e., no cycle is bicolored). | Adjacent vertices different; No cycle is bicolored. | Weaker than star coloring (a star coloring is always an acyclic coloring). |
A star coloring is always an acyclic coloring, meaning it also ensures that no cycle in the graph is bicolored (uses only two alternating colors). However, the converse is not true; an acyclic coloring is not necessarily a star coloring.
Applications of Star Coloring
While more complex to compute, star coloring finds applications in various fields where strict separation of elements is crucial:
- Frequency Assignment: In wireless networks, star coloring can help assign frequencies to transmitters to prevent interference, especially over short distances.
- Scheduling: It can be used in scheduling problems to ensure that tasks that are indirectly related (e.g., requiring shared resources with a mediating step) do not conflict.
- Data Analysis: In certain data clustering or network analysis scenarios, it might be used to group related but not directly conflicting elements.
Examples
Let's look at simple examples to illustrate star coloring:
-
Path Graph ($P_n$):
- For $P_2$ ($v_1-v_2$): Needs 2 colors (e.g., 1-2). $\chi_s(P_2) = 2$.
- For $P_3$ ($v_1-v_2-v_3$): Needs 3 colors (e.g., 1-2-3). Cannot be 1-2-1 because endpoints of $P_3$ must be different. $\chi_s(P_3) = 3$.
- For $P_4$ ($v_1-v_2-v_3-v_4$): Needs 3 colors (e.g., 1-2-3-1). Cannot be 1-2-3-2 because it would be a bicolored $P_4$.
- Generally, for $P_n$ where $n \ge 3$, $\chi_s(P_n) = 3$.
-
Cycle Graph ($C_n$):
- For $C_3$ (triangle): Needs 3 colors (e.g., 1-2-3). $\chi_s(C_3) = 3$.
- For $C_4$ (square): Needs 3 colors (e.g., 1-2-3-1). This is a valid star coloring. $\chi_s(C_4) = 3$.
- For $C_5$ (pentagon): Needs 4 colors (e.g., 1-2-3-4-1). This is because a $C_5$ has paths of length 3. For example, $v_1-v_2-v_3-v_4$. If $v_1=v_3$ or $v_2=v_4$ in a P4 sense. Or, if we use just 3 colors, say 1-2-3-1-2. Then $v_1-v_5-v_4$ is $1-2-1$, which is forbidden. So, $\chi_s(C_5)=4$.
