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Mỗi đồ thị trong Hình 5 có chu trình Euler không? Nếu có hãy chỉ ra một chu trình như vậy

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Đề bài

Mỗi đồ thị trong Hình 5 có chu trình Euler không? Nếu có hãy chỉ ra một chu trình như vậy. Nếu không, đồ thị có đường đi Euler không? Nếu có, hãy chỉ ra một đường đi như vậy.

Phương pháp giải - Xem chi tiết

Trong đồ thị, một đường đi được gọi là đường đi Euler nếu đường đi đó đi qua tất cả các cạnh của đồ thị, mỗi cạnh đúng 1 lần. Nếu chu trình là đường đi Euler thì chu trình đo được gọi là chu trình Euler.

Lời giải chi tiết

a) Đồ thị G:Ta có \(d\left( A \right){\rm{ }} = {\rm{ }}d\left( B \right){\rm{ }} = {\rm{ }}d\left( C \right){\rm{ }} = {\rm{ }}d\left( D \right){\rm{ }} = {\rm{ }}5.\)Suy ra 4 đỉnh của đồ thị G đều có bậc lẻ.Vậy đồ thị G không có chu trình Euler và cũng không có đường đi Euler.b) Đồ thị H:Ta có \(d\left( A \right){\rm{ }} = {\rm{ }}d\left( C \right){\rm{ }} = {\rm{ }}d\left( M \right){\rm{ }} = {\rm{ }}d\left( P \right){\rm{ }} = {\rm{ }}3;d\left( B \right){\rm{ }} = {\rm{ }}d\left( N \right){\rm{ }} = {\rm{ }}2.\)Suy ra đồ thị H có 4 đỉnh bậc lẻ.Vậy đồ thị H không có chu trình Euler và cũng không có đường đi Euler.

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