Ex - K-Coloring

AtCoder

IDabc294_h

Time8000ms

Memory256MB

Difficulty—

You are given a simple undirected graph with $N$ vertices numbered $1$ to $N$ and $M$ edges numbered $1$ to $M$. Edge $i$ connects vertex $u_i$ and vertex $v_i$. Find the number, modulo $998244353$, of ways to write an integer between $1$ and $K$, inclusive, on each vertex of this graph to satisfy the following condition: * two vertices connected by an edge always have different numbers written on them. ## Constraints * $1 \leq N \leq 30$ * $0 \leq M \leq \min \left(30, \frac{N(N-1)}{2} \right)$ * $1 \leq K \leq 10^9$ * $1 \leq u_i \lt v_i \leq N$ * The given graph is simple. ## Input The input is given from Standard Input in the following format: $N$ $M$ $K$ $u_1$ $v_1$ $u_2$ $v_2$ $\vdots$ $u_M$ $v_M$ [samples]

Samples

Input #1

Output #1

2

Here are the two ways to satisfy the condition.

*   Write $1$ on vertices $1, 3, 4$, and write $2$ on vertex $2$.
*   Write $1$ on vertex $2$, and write $2$ on vertex $1, 3, 4$.

Input #2

4 0 10

Output #2

10000

All $10^4$ ways satisfy the condition.

Input #3

Output #3

Input #4

Output #4

838338733

Input #5

7 12 1000000000
4 5
2 7
3 4
6 7
3 5
5 6
5 7
1 3
4 7
1 5
2 3
3 6

Output #5

418104233

API Response (JSON)

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      "content": "You are given a simple undirected graph with $N$ vertices numbered $1$ to $N$ and $M$ edges numbered $1$ to $M$. Edge $i$ connects vertex $u_i$ and vertex $v_i$.\nFind the number, modulo $998244353$, o...",
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Full JSON Raw Segments