⇐ Existence Proofs | Uniqueness Proofs ⇒ |

Claim:For allx≥ 0,^{⌈}x/2^{⌉}≥^{⌈}x^{⌉}/2 .

Proof:We break up the proof into three cases:

xis an even integer.xis an even integer plus someε,0 < ε < 1.xis an odd integer.xis an odd integer plus someε,0 < ε < 1.

Case 1:x = 2nfor some integern≥ 0.

Then,Also,^{⌈}x/2^{⌉}=^{⌈}2n/2^{⌉}=^{⌈}n^{⌉}=n.Thus,^{⌈}x^{⌉}/2 =^{⌈}2n^{⌉}/2 =2n/2=n.^{⌈}x/2^{⌉}≥^{⌈}x^{⌉}/2 .

Case 2:x = 2n + εfor some integern≥ 0 and some real numberεwhere 0 <ε< 1.

Then,We know^{⌈}x/2^{⌉}=^{⌈}(2n+ε)/2^{⌉}=^{⌈}n+ε/2^{⌉}=n+^{⌈}ε/2^{⌉}=n+ 1.^{⌈}ε/2^{⌉}= 1 because 0 <ε/2 < 1/2. Also,Again, we have^{⌈}x^{⌉}/2 =^{⌈}2n + ε^{⌉}/2 = (2n+^{⌈}ε^{⌉}) /2 = (2n+ 1) /2 =n+ 1/2.^{⌈}x/2^{⌉}≥^{⌈}x^{⌉}/2 .

Case 3:x= 2n+1 for some integern≥ 0.

Then,Also,^{⌈}x/2^{⌉}=^{⌈}(2n+ 1)/2^{⌉}=^{⌈}n+ 1/2^{⌉}=n+^{⌈}1/2^{⌉}=n+ 1.Thus,^{⌈}x^{⌉}/2 =^{⌈}2n+ 1^{⌉}/2 = (2n+ 1)/2 =n+ 1/2.^{⌈}x/2^{⌉}≥^{⌈}x^{⌉}/2 .

Case 4:x= 2n+ 1 +εfor some integern≥ 0 and some real numberεwhere 0 ≤ε< 1.

Then,We know^{⌈}x/2^{⌉}=^{⌈}(2n+ 1 +ε)/2^{⌉}=^{⌈}n+ 1/2 +ε/2^{⌉}=n+^{⌈}1/2 +ε/2^{⌉}=n+ 1.^{⌈}1/2 +ε/2^{⌉}= 1 becauseε< 1. Also,Thus, we have^{⌈}x^{⌉}/2 =^{⌈}2n+ 1 +ε^{⌉}/2 = (2n+ 1 +^{⌈}ε^{⌉}) /2 = (2n+ 2) /2 =n+ 1.^{⌈}x/2^{⌉}≥^{⌈}x^{⌉}/2 in the final case.QED

Claim:The graphKshown below is not a planar graph._{3,3}

Proof:Consider the verticesa,f,candd. They form a cycle in the graphK. Any planar drawing_{3,3}Kcannot have the edges cross, so the cycle divides the plane into two regions: inside and outside the cycle. Now, the vertex_{3,3}bcan either be placed inside or outside thea-f-c-d-acycle:

Case 1:Suppose that the vertexbis placed inside the cycle. Now, consider where the vertexecan be placed. There are 3 possible regions: inside the region bounded bya-f-b-d-a, inside the region bounded byc-d-b-f-cor outside thea-f-c-d-acycle.

In each of these possibilities, there is an edge that cannot be drawn without edges crossing.

Case 2:If the vertexbis place outside thea-f-c-d-acycle, we make the same argument about the placement of vertexe, except the roles of verticesaandbare interchanged.QED

⇐ Existence Proofs | Uniqueness Proofs ⇒ |

Last Modified: 30 Sep 2010 15:33:38 EDT by Richard Chang to Fall 2010 CMSC 203 Homepage