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A planar graph has faces (including the external face) composed of eight 3-cycles and eighteen 4- cycles. The graph is regular, that is the degree of all vertices are the same, and the same number of 4-cycles and 3-cycles meet at each vertex. (a) How many vertices and how many edges does it have? Explain. (b) How many 4-cycles and how many 3-cycles meet at each vertex? Explain. (Hint: it might help to think of 3-cycles as triangles and 4-cycles as squares.)

A planar graph has faces (including the external face) composed of eight 3-cycles and eighteen 4- cycles. The graph is regular, that is the degree of all vertices are the same, and the same number of 4-cycles and 3-cycles meet at each vertex. (a) How many vertices and how many edges does it have? Explain. (b) How many 4-cycles and how many 3-cycles meet at each vertex? Explain. (Hint: it might help to think of 3-cycles as triangles and 4-cycles as squares.)

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.3: Systems Of Inequalities
Problem 30E
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A planar graph has faces (including the external face) composed of eight 3-cycles and eighteen 4- cycles. The graph is regular, that is the degree of all vertices are the same, and the same number of 4-cycles and 3-cycles meet at each vertex. (a) How many vertices and how many edges does it have? Explain. (b) How many 4-cycles and how many 3-cycles meet at each vertex? Explain. (Hint: it might help to think of 3-cycles as triangles and 4-cycles as squares.)
Transcribed Image Text:A planar graph has faces (including the external face) composed of eight 3-cycles and eighteen 4- cycles. The graph is regular, that is the degree of all vertices are the same, and the same number of 4-cycles and 3-cycles meet at each vertex. (a) How many vertices and how many edges does it have? Explain. (b) How many 4-cycles and how many 3-cycles meet at each vertex? Explain. (Hint: it might help to think of 3-cycles as triangles and 4-cycles as squares.)
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