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Multilevel Distance Labeling - A Wireless Network Problem Tian-Shun Allan Jiang ABSTRACT Multilevel distance labeling is a graph-theoretical solution to the problem of frequency assignment on wireless networks. An optimal labeling reduces the range of radio frequencies assigned to radio stations and eliminates network interference. Due to the ubiquity of wireless networks, a more efective frequency assignment is an important area of study to increase eiciency and quality of communication. We model the problem by representing broadcasting stations as vertices on a graph. A radio labeling of a connected graph G is a mapping F:V(G)→{0,1,2,…} such that |F(u)-F(v)|+d(u,v)≥diam(G)+1 for each pair of distinct vertices u,v∈V(G) where diam(G) is the diameter of G and d(u,v) is the distance between u and v. he span of F, denoted span(F), is deined max(u,v ∈ V(G)) |F(u)F(v)|. hen the radio number of G is denoted In this paper, we introduce a general method to compute the lower bound for rn(G), introduce a method to characterize solutions F on G, and prove a closed-form formula of rn(G) for the path and triangle lollipop graphs.
Introduction Background Wireless communication pervades modern society. Wireless internet, mobile phones, radio, and GPS are just a few of the common applications of wireless technology. An eicient and reliable wireless network must overcome a number of technical challenges; among these is the allocation of broadcast frequencies to minimize interference. An efective method of frequency coordination, a regulatory process for the mitigation of frequency interference, increases the eiciency of wireless communication. In this paper, we focus on the problem of frequency coordination in cellular networks. Cellular systems are designed to minimize both interference and range of channel assignment through frequency reuse. In this system, the coverage area is partitioned into many cells with assigned frequencies. Since signal power is efective within a certain radius from the transmitter, reuse of similar frequency spectra becomes possible at certain distances [4]. his reuse allows cellular system designers to minimize the frequency range used for the whole system. he distance among cells that use similar frequency spectra should be minimized to increase spectral eiciency. However, if the distance is too small, users will receive frequencies from both channels, causing intercell interference [4]. hus, a balance between spectral eficiency and inter-cell interference should be achieved. In this paper, we present a graph-theoretical model of a solution which eliminates inter-cell interference while maximizing spectral eiciency.
Multilevel Distance Labelling We represent cellular stations with vertices on a graph G, and draw edges between vertices if the stations are geographically close. Interference among stations can occur at multiple levels, ranging from the interference between the closest stations with distance one, to the furthest stations, with distance diam(G). Given a connected graph G, for two vertices u,v∈V(G) let d(u,v) be the distance between u and v. hen a radio labeling of G is a function F: V(G)→{1,2,3,…} such that for u,v∈V(G):
he span of F is, span = max(u,v ∈ V(G)) |F(u)-F(v)|. he radio number of G, denoted rn}(G):=min(span(F)). he solutions of G are all labelings F such that span(F)=rn(G). In past work on the problem [1], the usual method has been to ind an upper bound of rn(G) equal to the lower bound of rn(G). However, neither the upper nor lower bounds are easy to establish. his paper addresses some of the challenges and insights when searching for rn(G). In Section 2, some preliminary investigations of the distance labeling problem are presented, and in Section 3, we establish a general methodology to ind the lower bound of a graph G. In Sections 5 and 6, we focus on inding the upper and lower bounds of the radio numbers of two speciic classes of graphs: path and lollipop. his inds and proves the closed form expression of rn(G) for these two graph types. In Section 7, we introduce tightness graphs as a way to classify solutions and in Section 8, several areas of further research are discussed.
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Street Broad Scientific Preliminaries Deinitions 1. Wiggle Room: For two vertices x and y, deine the wiggle room wr(x,y)=|F(x)-F(y)|+d(x,y)-(D+1). Notice that in a valid distance labeling, the wiggle room is nonnegative. 2. Tight: Two vertices x and y are called tight if wr(x,y)=0. 3. Tightness Graphs: he tightness graph GT of a labeled graph G has vertex set V(GT)=V(G) and edge set E(GT )={(x,y)| x,y ∈V(G), wr(x,y)=0}. his idea is further explored in Section 7. 4. Hopping: We deine a hopping, or hopping sequence H(G) to be a permutation of V(G) such that H(G)={h1, h2,…,hn} and F(hi) < F(h(i+1)) for 1 ≤ i ≤ n-1. 5. Tight Hopping: A special case of hopping is where wr(hi, hi+1) = 0 for all 1 ≤ i ≤ n-1. hese are referred to as tight hoppings. Terminology 1. A graph G is deined as G=(V(G),E(G)), where V(G) is the vertex set of G, E(G) is edge set of G, and an edge e∈E(G) is a subset of two vertices v∈V(G). 2. he distance d(x,y) for x,y∈V(G) is the length of a shortest path between x and y. 3. he diameter diam(G) or D of a graph is the maximum distance between two vertices v∈V(G). 4. he path graph Pn is a graph with vertices V(Pn)={v1, v2,…, vn} and edges E(Pn)={(v1, v2),(v2, v3),…, (v(n-1), vn)}.
REsEaRch Observations 1. Do Not Repeat Labels: Claim: No two vertices can have the same labeling. Proof by Contradiction: Assume that there exist two vertices x,y such that F(x)=F(y). By deinition, d(x,y)+|F(x)F(y)| > D and d(x,y) > D. However, this is a contradiction, as d(x,y) ≤ D. 2. An Obvious Upper Bound: Claim: rn(G)≤(n-1)∙D. Proof by Construction: Let our labeling F:V(G)→{0, D, 2D,…,(n-1)D}. his is a distance labeling, because |F(x)F(y)| ≥ D and d(x,y) ≥ 1, which complies with the distance labeling deinition that d(x,y)+|F(x)-F(y)| ≥ D+1. 3. he Inverse Solution: Claim: Given a solution F of G, there exists a corresponding solution F’ of G. Proof by Construction: Let F’(vi)=rn(G)-F(vi). he assignment F’ is also valid, as d(x,y)+|(rn(G)-F(x))(rn(G)-F(y))|=d(x,y)+|F(x)-F(y)|>diam(G). Furthermore, we have span(F)=span(F’). We call this equivalent solution F’ the inverse solution of F. Below is an example of an inverse solutions S1 and S2.
Figure 2.2.1. Inverse Solutions on P5 Figure 2.1.1. Example Path Graph 5. A triangle lollipop graph TLn is a graph with vertices V(TLn)=V(Pn) and edges E(TLn)=E(Pn)U(vn,v(n-2)). For sake of convenience, let us call vn the “lollipopped” vertex.
Figure 2.1.2. Example Triangle Lollipop Graph
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4. A Flawed Labeling: In initial investigations of multidistance labeling on Pn, the following labeling algorithm was conceived: Let us deine a permutation P of V(G) such that
and Pi is the ith element in the permutation. Let F(P0)=0. hen, for all i, label F(P(i+1)) such that wr(F(Pi), F(P(i+1)))=0 (see Deinition 2.1.1), and F(Pi)<F(P(i+1)) for 1 ≤ i ≤ [(n+1)/2], and F(Pi) > F(P(i+1)) for [(n+1)/2] ≤ i ≤ n. (S1 in Figure 2.2.1 is an example of this algorithm.)
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REsEaRch It is easily veriied that this labeling is valid. We also see that span(F)=F(v([(n+1)/2]+1) ). After some calculation, we see that:
Although this algorithm gives rn(G) for all paths with at most 6 vertices, it no longer works for 7 or more vertices, as it was disproven by the computer (see Appendix B problems.). However, this labeling is insightful in that it recognizes that rn(G) grows approximately as (n2), indicating that rn(G) is likely a quadratic function. With these observations and deinitions, we have some interesting tools to approach the distance labeling problem. Of particular interest is the notion of tight hopping. Some experimentation will reveal that not all tight hopping sequences lead to valid distance labelings. Below, we establish a lemma that places proper restrictions on hopping to ensure that it generates a valid labeling for the case of a path graph Pn and triangle lollipop graph TLn .
Distance Labeling on Paths and Triangle Lollipops
We prove that this rule will make the tight hopping on a path a valid distance labeling, by showing that this labeling satisies the deinition |F(u) - F(v)| ≥ diam(G) - d(u,v) + 1 for any two vertices u and v. First, we note that any two vertices hi and hj with |i-j|=1 will satisfy the relationship, as wr(hi, hj)=0. Further, any two vertices hi and hj with |i-j|=2 satisfy the deinition. Let d(hi, h(i+1))=d1 and d(h(i+1), h(i+2))=d2. Now, we may express d(hi, h(i+2))=d3 in terms of d1 and d2. First, we establish the value of h(i+2) compared to hi.
From these equations we get F(h(i+2))-F(hi)=2D+2-d1d2. Now there are two cases: Case 1: d3 = d1 + d2
In this section, we establish some rules and restrictions on tight hoppings for Pn and TLn to ensure that the resultant labeling is a valid distance labeling. Tight Hopping Rules on Paths and Lollipops he following restrictions are necessary and suicient to ensure that a tight hopping on a path or lollipop is a valid distance labeling: 1. If our hopping sequence contains hi→h(i+1)→h(i+2), such that d(hi, h(i+1)) > d(h(i+1), h(i+2)) and d(xi, x(i+2))=|d(hi, h(i+1))-d(h(i+1), h(i+2))|, then d(h(i+1), h(i+2)) ≤ (D+1)/2. 2. If our hopping sequence contains hi→h(i+1)→h(i+2), such that d(hi, h(i+1)) < d(h(i+1), h(i+2)) and d(xi, x(i+2))=|d(hi, h(i+1))-d(h(i+1), h(i+2))|, then d(hi, h(i+1)) ≤ (D+1)/2. Notice that if our hopping sequence is hi→h(i+1)→h(i+2), with d(xi, x(i+2))=d(hi, h(i+1))+d(h(i+1), h(i+2))|, then min(d(hi,h(i+1)), d(h(i+1), h(i+2))) ≤ (D+1)/2 - as we cannot partition a path of length D+1 into two parts with length greater than (D+1)/2. With this, we notice that the restrictions placed on tight hoppings are equivalent to the following restriction:
Clearly, if two hops are in the same direction, we have d3=d1+d2. hen, by the distance labeling deinition, F(h(i+2))-F(hi)≥D+1-d3. Since we already know F(h(i+2))-F(hi)=2D+2-d1-d2, we have
Since D+1>0, when hopping twice on a path in the same direction, there are no restrictions on the values of d1 and d2, other than d1+d2 ≤ D+1 → min(d1, d2) ≤ (D+1)/2 as desired. Case 2: d3 = |d1+d2 | In this case, the two hops are in opposite directions. Without loss of generality, we may let d1>d2. hen, by deinition we have, F(h(i+2))-F(hi) ≥ D+1-d3. Again, since F(h(i+2))F(hi)=2D+2-d1-d2, we have
Since we let d1>d2., we have D+1 ≥ 2d2→(D+1)/2 ≥ d2 as desired. For all cases of vertices hi and hj with i-j≥3, we prove that Volume 2 | 2012-2013 | 19
Street Broad Scientific our restrictions on tight hopping satisfy the deinition through induction. We already have two base cases: i-j=1 and i-j=2 from above. Now, assume that all pairs of vertices hi, hj with i-j ≤ k have wr(hi, hj) ≥ 0. hen, if we consider the hopping sequence hi→h(i+1)→∙∙∙→h(i+k)→h(i+(k+1)), we see that wr(hi, h(i+k)) ≥ 0, and wr(h(i+k), h(i+k+1)) ≥ 0. hus, the vertices hi, h(i+k), h(i+(k+1)) satisfy our above restriction on hopping. hen, we have wr(hi,h(i+(k+1))) ≥ 0. As this proves our induction step, it follows that the a tight hopping with the restriction d(hi,h(i+1)) ≤ (D+1)/2 generates a valid distance labeling on Pn Verifying Path Labelings Above, we showed that any tight hopping on a path or lollipop satisfying:
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Since rn(G) ≥ min(F(xn)), if we can maximize ∑(i=1)(n-1) di on a graph G, we will have a lower bound for rn(G). We call ∑(i=1)(n-1)di the total hopping distance, as it is the sum of all the distances as we hop over a sequence of the n vertices of G. hen we see that our lower bound makes sense, as increasing the total hopping distance decreases the amount we must increment the labelings. his total hopping distance can be maximized for the path graph, and the triangle lollipop graph. However, for a general graph G, inding the maximum total hopping distance may be NP-complete due to a reduction from L(2,1) labelings [3].
Proof for Paths will be a valid distance labeling. hus, one way to show that a given labeling F is indeed a valid distance labeling on Pn or TLn, is by showing that: 1. he labeling is a tight hopping, and 2. For 1 ≤ i ≤ n-2, we have min(d(hi, h(i+1)), d(h(i+1), h(i+2))) ≤ n/2.
here are certain classes of graphs for which we may compute the lower bound and construct an upper bound matching the lower bound. his ability to compute the lower bound is related to the ability to ind the maximum hopping distance. It follows that paths are special cases when trying to ind the maximum hopping distance, as distances are found by subtracting vertex numbers.
he Lower Bound
heorem: For any n ≥ 4
We describe a method to establish a lower bound for graph G. Total Hopping Distance Let us consider the vertices of G in order of increasing label. hen, if G has vertices V(G)={v1,v2,…,vn}, let {x1, x2,…, xn}, be a permutation of V(G) such that F(x(i+1))>F(xi) for all 1 ≤ i ≤ n-1.
We prove the result by sandwiching the value of rn(G) between a coinciding upper bound and lower bound.
For convenience, let F(x(i+1))-F(xi)=fi and d(x(i+1), xi) = di.
Lower Bound From Section 3, we know that F(xn) ≥ (n-1)(D+1)-∑(i=1) (n-1)di It happens that the maximum hopping distance is diferent for even length and odd length paths.
By deinition we have F(x1) ≥ 0 and fi ≥ D+1-di. We also deine the contribution of a vertex cb(xi)=fi+di-D-1. We see that fi is minimized when cb(xi)=0 for 1 ≤ i ≤ n-1.
Odd Length Paths Note: if we have ∑(i=1)((2k+1)-1)di ≤ 2k2+2k-2 then we are done, as:
Now, we note that the maximum labeled vertex F(xn)=∑(i=1)(n-1)fi . here exists an assignment of di’s such that F is a valid distance labeling, so we have
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Claim: if ∑(i=1)2kdi > 2k2+2k-2 then ∑(i=1)^2kdi =2k2+2k-1, and there exists a vertex vi such that wr(vi)=1
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REsEaRch Proof of Claim: Since each d(xi, x(i+1))=|j-j’| if xi = vj and x(i+1)=vj’, ∑(i=1)2kdi is the sum of 4k j’s where half of the j’s are positive, the other half are negative, and 1 ≤ j ≤ 2k+1. Furthermore, 2k-1 terms appear twice, and 2 terms appear once. (he terms appearing once represent the min and max labeled vertex).
In Case 2, we have v(k+1)=x1 and v(k+2)=x(2k+1). Like the previous case, we know that if xi ≥ k+1, then x(i+1) ≤ k. Now, consider xi=2k+1. hen, x(i-1) ≤ k and x(i+1) ≤ k. his again contradicts our hopping criterion above. As both distances di-1, di ≥ (D+1)/2, we know that either cb(x(i-1))=1 or cb(x(i+1))=1, which forces our maximum value to increment by at least 1. By taking the sum of all inequalities and accounting for the contribution of 1, we have
and we have shown the lower bound for odd paths. Figure 5.1.1. Assigning values of j on P5 hen, to maximize ∑(i=1)2kdi, we need to minimize the absolute values of the negative terms and maximize the values of the positive terms. here are two cases achieving this maximum summation: Case 1: We have positive values of j belonging to {k+2, k+3,…, 2k+1}, each of which appears twice (note that this is 2k terms), negative values of j belonging to {1, 2,…, k-1}, each of which appears twice, and negative values for k and k+1 both appearing once. Case 2: We have positive values of j belonging to {k+3, k+4,…, 2k+1}, }, each of which appears twice, positive values for k+1 and k+2, and negative values of j belonging to {1, 2,…, k}, each of which appears twice. In both cases, we get:
In Case 1, we have v(k+1)=x1 and vk=x(2k+1). (It does not matter if the positions of x1 and x(2k+1) are switched due to the inverse solution). Since each di is composed of a positive component and a negative component, we know that if xi ≥ k+2 (in part B below), then x(i+1) ≤ k+1.
Even Length Paths Finding this lower bound is simpler than for odd length paths, as there is only one way to maximize the hopping distance. his is due to the even split between positive and negative terms of j. Claim: ∑(i=1)(2k-1)di ≤ 2k2-1 Proof of Claim: Using the above logic, we know that we have 4k-2 terms of 1 ≤ j ≤ 2k, 2k-2 of which occur twice and 2 of which occur once. Again, half of these terms are positive and the other half are negative. he maximization of this sum occurs when we have positive values of j belonging to {k+2, k+3,…, 2k}, each of which appears twice, positive values for k and k+1 each appearing once, and negative values of j belonging to {1, 2,…, k-1}, each of which appears twice. From this, we get:
If follows that:
as desired [1]. Upper Bound Odd Length Paths Now, consider xi=1. hen, x(i-1) ≥ k+2 and x(i+1) ≥ k+2 . However, this contradicts our tight hopping criterion from Section 3.1. hus, since both distances di-1, di ≥ n/2, we know that either cb(x(i-1))=1 or cb(x(i+1))=1, which forces our maximum value to increment by at least 1. Volume 2 | 2012-2013 | 21
Street Broad Scientific We label the vertices of G as follows:
REsEaRch hen we get x(2k+1)=2k(2k+1)-∑(i=1)2kdi , and the problem is reduced to inding the hopping distance of this speciic labeling scheme. However, inding the hopping distance of a labeling algorithm is not so diicult, and algebra veriies that ∑(i=1)2kdi =2k^2+2k-2, giving x(2k+1)=(k+1)2+(k-1)2. Even Length Paths We label the vertices of G as follows:
Table 5.2.1. Labelling Algorithm for P2k+1 where x1=0, and each xi is tight with x(i+1) for 1 ≤ i ≤ n-1. We proceed to show that this tight hopping is a distance labeling by checking it against the restrictions placed in Section 3.1. Note: his labeling only works for odd paths with n ≥ 7. In the case of n=5, x(2k+1) is too great with the above labeling. However, the case n=5 does follow the formula given above for rn(P5). Furthermore, all solutions generated with the above labeling have GT which are also path graphs. (See Section 7.2 for examples). Proof of Upper Bound Using our distance labeling veriication method for paths from Section 2.6.1 and Section 2.6.2, we need min(di, d(i+1)) ≤ n/2. Checking the labeling above, for every three consecutive vertices, at least one adjacent pair is k apart. As k < n/2, we see that the above labeling method is valid. Now, we need to show that the above method achieves the upper bound x(2k+1)=(k+1)2+(k-1)2. Since this is a tight hopping, there are no contributions from any vertices. herefore:
Table 5.2.2. Labeling Algorithm for P2k where x1=0, and each xi is tight with x(i+1) for 1 ≤ i ≤ n-1. We again check against the restrictions placed in Section 3.1 Proof of Upper Bound Once again, we need min(di, d(i+1)) ≤ n/2. Checking the labeling above, for every three consecutive vertices, at least one adjacent pair is k apart. As k < n/2, we see that the above labeling method is valid. Now, we need to show that the above method achieves the upper bound x2k=(k-1)2+k2. Since this is a tight hopping, there are no contributions from any vertices. herefore:
hen we get x2k=2k(2k-1)-∑(i=1)(2k-1)di. It is easy to verify that ∑(i=1)^(2k-1)di =2k2-1, giving x2k=(k-1)2+k2 22 | 2012-2013 | Volume 2
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As from Section 5.1 we got the lower bound of rn(Pn) for even and odd n, and have constructed cases achieving these lower bounds, thereby completing the proof to path graphs.
To maximize ∑(i=1)2kdi , we minimize the absolute values of the negative terms and maximize the values of the positive terms. here are two cases achieving this maximum summation:
Proof for Triangle Lollipops
Case 1: We have positive values of j belonging to {k+2, k+3,…, 2k-1, 2k, 2k}, each of which appears twice (note that this is 2k terms), negative values of j belonging to {1, 2,…, k-1}, each of which appears twice, and negative values for k and k+1 both appearing once.
heorem:
Lower Bound We continue to use the maximum hopping distance technique. Once again, there is a diferent maximum hopping distance for even and odd lollipops. Odd Length Lollipops Since:
if ∑(i=1)2kdi ≤ 2k2+2k-3, we have the desired lower bound. Claim: ∑(i=1)2kdi ≤ 2k2+2k-3 Proof of Claim: We use the property that d(xi, x(i+1))=|jj’| if xi=vj and x(i+1)=vj’ (and when xi=v(2k+1), j=±2k) for all distances except for d(v(2k+1), v2k) =1, as shown in Figure 6.1.1. However, we may show that∑(i=1)2kdi does not include d(v(2k+1), v2k).
Case 2: We have positive values of j belonging to {k+3, k+4,…, 2k-1, 2k, 2k}, each of which appears twice, positive values for k+1 and k+2, and negative values of j belonging to {1, 2,…, k}, each of which appears twice. In both cases, we get ∑(i=1)2kdi =2k^2+2k-3 as desired. Even Length Lollipops his case is more diicult than the odd length lollipop, an observation that is pretty intuitive that this case is more complicated after completing the proofs for the path graphs. Claim: ∑(i=1)2kdi ≤ 2k2-3 with 2 distinct values of i such that cb(xi)=1. Proof of Claim: Using the above logic, we know that we have 4k-2 terms of 1 ≤ j ≤ 2k, with k-1 positive terms appearing twice, k-1 negative terms appearing twice, and 1 positive and 1 negative term appearing once. he maximization of this sum occurs when we have positive values of j belonging to {k+1, k+2,…, 2k, 2k-1 , 2k-1}, each of which appears twice, a positive k appearing once, and negative values of j belonging to {1, 2,…, k-1}, each of which appears twice, and a negative k+1 appearing once. From this, we get:
Figure 6.1.1. Assigning Values of j on TL5 Suppose that ∑(i=1)2kdi does include d(v(2k+1), v2k). We notice that the remaining distances of the triangle lollipop are in fact equivalent to P2n. Since the maximum hopping distance of P2n was 2k2-1 we have ∑(i=1)2kdi=(2k21)+1=2k2. We proceed to construct a hopping with a greater distance. Since we have 2k+1 vertices, and 2 vertices are designated to be x1 and x(2k+1)}, we have (2k+1-2)∙2 +2 terms of j. hus, our sum has 4k terms, with 2k positive and 2k negative.
Now, we prove that the contribution to the maximum value is 2. Let the positive terms of j in our summation occur in region B, where j ≥ k+1, and the negative terms
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appear in region A, where j ≤ k.
Section 3.1
Figure 6.1.2. Calculating Contributions on TL2k
Note: his labeling only works for odd lollipops with n ≥ 7. In the case of n=5, the algorithm given above fails. However, the case n=5 does follow the formula given above. (run the code in Appendix B to Furthermore, all solutions generated by the algorithm have a consistent structure in GT.
Without loss of generality, let x1=vk. Now, if we consider xi=v1 then x(i-1) and x(i+1) both occur in region B. his gives min(di, d(i+1)) > (D+1)/2cb(x(i+1)) ≥ 1. In particular, cb(x(i+1))= min(di, d(i+1)) - (D+1 )/2 for min(di, d(i+1)) ≥ (D+1 )/2. hus, if min(di,d(i+1)) ≥ k+1, then, cb(x(i+1)) ≥ 2 and we are guaranteed the desired contribution of 2 to the maximum value. However, if min(di, d(i+1))=k→ x(i+1)=v(k+1), then cb(x(i+1))=1. In this case, we need to ind another pair of distances where both are at least k. Consider xj=v2. hen, to minimize cb(x(j+1)) we have xj=v(k+2). his again gives us cb(x(j+1))=1, so we have the two contributions to the maximum. Our current lower bound is rn(G) ≥ (2k-1)(2k-2+1)(2k2-3)+2= 2k2-4k+6.
Proof of Upper Bound Using our distance labeling veriication method for paths from Section 3.1, we need min(di, d(i+1)) ≤ (D+1)/2. Checking the labeling above, for every three consecutive vertices, at least one adjacent pair is k-1 apart. As k-1 < (D+1)/2, we see that the above labeling method is valid. Now, we need to show that the above method achieves the upper bound x(2k+1)=2k2-2k+3. Since this is a tight hopping, there are no contributions from any vertices. herefore:
hen we get x2k=2k(2k-1)-∑(i=1)(2k-1)di. It is easy to verify that ∑(i=1)(2k-1)di=2k2-1, giving x2k=(k-1)2+k2
Upper Bound Odd Length Lollipops We label the vertices of G as follows:
As from Section 5.1 we got the lower bound of rn(Pn) for even and odd n, and have constructed cases achieving these lower bounds, thereby completing the proof to path graphs. Now, we need to show that the above method achieves the upper bound x(2k+1)=2k2-2k+3. Since this is a tight hopping, there are no contributions from any vertices. herefore, x(2k+1)=∑(i=1)2k D+1-di. hen we get x(2k+1)=(2k+1-1)(2k-1+1)-∑(i=1)2kdi, and once again it comes down to ind the hopping distance of this speciic labeling scheme. However, inding the hopping distance of a labeling algorithm is not so diicult, and some algebra veriies that ∑(i=1)2kdi=2k2+2k-3, giving x(2k+1)=2k2-2k+3 Even Length Lollipops We label the vertices of G as follows:
Table 6.2.1. Labeling Algorithm for TL2k+1 where x1=0, and each xi is tight with x(i+1) for 1 ≤ i ≤ n-1. Similarly, we check against the restrictions placed in 24 | 2012-2013 | Volume 2
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REsEaRch Tightness Graphs
Introduction Tightness graphs, GT, are an interesting way to categorize solutions, as they reveal the underlying structure of a solution. One nice application of the tightness graph is its use in generating a labeling algorithm for new graph types. his could help generate an upper bound for the graph type in general. Using the Python code in Appendix B, we may view GT for several Pn. hen, we construct a labeling algorithm by comparing solutions with similar GT. he next subsection features several structures and examples of solutions and their corresponding GT. Structures and Examples Solutions of P2k+1 with graphs GT that are also paths:
Table 6.2.2. Labeling Algorithm for TL2k where x1=0, and each xi is tight with x(i+1) for 1 ≤ i ≤ n-1. Again, we check against the restrictions placed in Section 3.1 Note: his labeling algorithm only works for n ≥ 8. All cases TL2k with k < 4 were computed by the computer. Proof of Upper Bound We need min(di, d(i+1)) ≤ (D+1)/2. Checking the labeling above, for every three consecutive vertices, at least one adjacent pair is k-1 apart, except in those distance pairs where cb(x(i+1)) > 0. As indicated in the section about the lower bound, it is precisely vertices v1 and v2 for which cb(vi)=1. Now, we need to show that the above method achieves the upper bound x2k=2k2-4k+6. Since we have ∑(i=1) kdi=2k2-3, x(2k+1)=(2k-1)(2k-2+1)-(2k-3)+2. Some algebra reveals that ∑(i=1)2kdi=2k2-4k+6 as desired. So we have found rn(G) for the lollipop graph.
Figure 7.2.1. Graphs of solutions to P2k+1 and associated GT
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Street Broad Scientific Observe that all of these solutions follow the labeling algorithm in Section 5.2.1
REsEaRch From this section, we conclude that inspecting the structures of GT is a quick way to identify and organize solutions on G.
Conclusion and Future Work
Figure 7.2.2. Solutions of P2k with graphs GT with a ladder structure
We have been able to establish a general method to inding the lower bound of any graph G using the idea of the maximum hopping distance. In Sections 5 and 6, we were able to ind these lower bounds for Pn and TLn and construct upper bounds that matched these values. he key contribution of this work is that it demonstrates a method by which rn(G) can be found and proven on a given class of graph. Additionally, we were able to use computer simulation to ind solutions to any new types of graph, and understand characteristics of these solutions. Further work is underway to ind rn(G) of the triangular lattice graph. hese graphs are of particular interest, because cellular systems generally have broadcasters located in the pattern of a triangular lattice. hus, determining the optimal frequency coordination of such a graph will be of greater application than for path graphs or lollipop graphs.
Acknowledgements I would like to thank Dr. Teague for his help and support of my work during this project. I was able to bounce a lot of ideas of of him, and he really helped me stay hopeful when it seemed as though my ideas had dried up.
References Figure 7.2.3. Graphs of solutions to P2k and associated graphs GT Notice that all of these solutions follow the labeling algorithm in Section 5.2.2. However, not all solutions to Pn have similar GT structures. Figure 7.2.3 shows two examples of alternative structures for even and odd length paths. Notice that they have diferent labeling algorithms from the examples in Figures 7.2.1 and 7.2.2. Instead, solutions with tightness graphs following structures from Figure 7.2.3 have separate labeling algorithms.
Figure 7.2.4. Graphs of solutions to P2k and associated graphs GT 26 | 2012-2013 | Volume 2
[1] D. Liu and X. Zhu, Multi-level distance labelings for paths and cycles, SIAM J. Disc. Math., in press, 2006. [2] J. A. Gallian, A dynamic survey of graph labeling, Electronic J. of Combinatorics, DS NO. 06, 16(2009). [3] J. R. Griggs, R.K. Yeh, Labeling graphs with a condition at distance two, SIAM J. Discrete Math. 5 (1992), 586â&#x20AC;&#x201C;595. [4] Molisch Andreas, Wireless Communications, WileyIEEE, New York, 2005.