Question
Number of possible distinct graphs G with the given vertex-distance condition
Original question: Let be a set of 10 distinct vertices labeled . Let be a simple graph representing a tunnel network, such that is 4-regular (i.e., every vertex has a degree of 4). Consider a set of 10 individuals, where each individual is assigned an integer age from the set . Individuals with the same age are defined as incompatible. It is given that there exists a bijective mapping from the set of individuals to the vertices such that for any pair of incompatible individuals, the shortest path distance between their assigned vertices in is at least 3 (or infinity, if they lie in different connected components). Compute the remainder when the number of possible distinct graphs (specifically, the number of distinct edge sets ) that satisfy these conditions is divided by 1000.
Expert Verified Solution
Key takeaway: This is a graph theory counting problem, but the statement omits key data needed for a unique numerical answer. The main issue is that the age distribution of the 10 individuals is not specified, so the incompatibility graph is not fully determined.
As written, this question does not determine a unique remainder.
Why the answer is not well-defined
You are asked to count the number of distinct 4-regular simple graphs on 10 labeled vertices such that there exists a bijection from 10 individuals to the vertices and every pair of individuals with the same age is mapped to vertices at distance at least 3.
However, the statement is missing essential information:
- it does not say how many individuals have each age,
- it does not specify whether the age assignment is fixed or variable,
- it does not define whether the graph is counted up to isomorphism or by labeled edge set in a way that makes the compatibility condition independent of the age multiplicities.
Because the number of same-age pairs depends on the age distribution, the set of graphs that satisfy the condition also changes. So the count cannot be uniquely determined from the information given.
What can be concluded
- If all 10 individuals had distinct ages, the incompatibility condition would be vacuous, and the problem would reduce to counting all 4-regular graphs on 10 labeled vertices.
- If some ages repeat, the distance constraint becomes nontrivial and removes some graphs, but the exact number depends on the missing age counts.
Therefore, the only mathematically sound response is:
Pitfalls the pros know 👇 A frequent mistake is to force a numerical answer even when the data needed for counting are incomplete. In graph counting problems, the exact multiplicities of constraints matter. Here, without the age distribution, there is no single correct count to reduce modulo 1000.
What if the problem changes? If the ages were specified, one could proceed by first determining which vertex pairs must be at distance at least 3, then counting labeled 4-regular graphs that satisfy those forbidden-distance constraints. If all ages were distinct, the problem would collapse to counting all 4-regular graphs on 10 labeled vertices.
Tags: 4-regular graph, graph distance, counting problem
FAQ
Why does this graph counting problem have no unique numerical answer?
Because the statement does not specify how many individuals have each age. The number of same-age pairs changes the constraint, so the set of valid graphs changes as well.
What extra information is needed to solve it?
You would need the exact age distribution of the 10 individuals, and ideally a clear statement about whether graphs are counted as labeled edge sets or up to isomorphism.