DEBRUIJN-LIKE SEQUENCES AND THE IRREGULAR CHROMATIC NUMBER OF PATHS AND CYCLES

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1 DEBRUIJN-LIKE SEQUENCES AND THE IRREGULAR CHROMATIC NUMBER OF PATHS AND CYCLES MICHAEL FERRARA, CHRISTINE LEE, PHIL WALLIS DEPARTMENT OF MATHEMATICAL AND STATISTICAL SCIENCES UNIVERSITY OF COLORADO DENVER ELLEN GETHNER DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF COLORADO DENVER Astract. A ebruijn sequence of orer, or a -ebruijn sequence, over an alphaet A is a sequence of length A in which the last element is consiere ajacent to the first an every possile -tuple from A appears exactly once as a string of -consecutive elements in the sequence. We will say that a cyclic sequence is ebruijn-lie if for some, each of the consecutive -element sustrings is istinct. A vertex coloring χ : V (G) [] of a graph G is sai to e proper if no pair of ajacent vertices in G receive the same color. Let C(v; χ) enote the multiset of colors assigne y a coloring χ to the neighors of vertex v. A proper coloring χ of G is irregular if χ(u) = χ(v) implies that C(u; χ) C(v; χ). The minimum numer of colors neee to irregularly color G is calle the irregular chromatic numer of G. The notion of the irregular chromatic numer pairs nicely with other parameters aime at istinguishing the vertices of a graph. In this paper, we emonstrate a connection etween the irregular chromatic numer of cycles an the existence of certain ebruijn-lie sequences. We then etermine exactly the irregular chromatic numer of C n an P n for n 3, thus verifying two conjectures given y Oamoto, Racliffe an Zhang. Keywors: ebruijn Sequence, Irregular Chromatic Numer, Distinguishing 1. Introuction 1.1. Terminology an Notation. In this paper we consier only simple graphs, namely those graphs with no loops an no multiple eges. We allow P n an C n to enote the path an cycle of orer n, respectively. Aitionally, [] will enote the set {1,...,} an if D is a igraph A(D) will signify the arc set of D. Let n (G) enote the numer of vertices of a graph G with egree. For any unefine terms, we refer the reaer to [9]. A -coloring of a graph G is an assignment of laels ( colors ) from the set [] to the vertices of G. A vertex coloring χ : V (G) [] of a graph G is sai to e proper if no pair of ajacent vertices in G receive the same color. The chromatic numer of a graph G, enote χ(g), is the smallest such that there exists a proper -coloring of G.

2 2 M. FERRARA, E. GETHNER, C. LEE, P. WALLIS 1.2. DeBruijn-Lie Sequences. A ebruijn sequence of orer, or a -ebruijn sequence, over an alphaet A is a sequence of length A in which the last element is consiere ajacent to the first an every possile -tuple from A appears exactly once as a string of -consecutive elements in the sequence. We will say that a cyclic sequence is ebruijn-lie if for some, each of the consecutive -element sustrings is istinct. In part ue to the connection etween ebruijn sequences an DNA sequencing, the prolem of fining ebruijn-lie sequences with aitional structure has receive an increase amount of attention recently. A goo example of this is [4] in which the authors escrie a graphical approach to generating ebruijn-lie sequences where each consecutive -element sustring is istinct up to reversal an complementation (in the same way that the DNA ase pairs C-G an A-T are complementary). In this paper, we construct ebruijn-lie sequences that allow us to approach a specific prolem in graph coloring, escrie in the next section. DeBruijn-lie sequences (of any type) are also of interest as purely cominatorial ojects, inepenent of their various applications Irregular Vertex Colorings. The prolem of eveloping a metho y which one can istinguish the vertices of a graph has een of interest for some time. Most examples of such methos generally involve laeling the eges or vertices of a graph in a manner that allows for ifferentiation of aritrary pairs of vertices. Some examples inclue (ut are certainly not limite to) the irregularity strength of a graph, first introuce in [10], istance coes [13, 18] an vertex istinguishing ege colorings, introuce in [7]. The following metho for ifferentiating vertices is of interest here. In [2], the authors efine a istinguishing coloring of a graph G to e a (not necessarily proper) coloring of the vertices of G that is preserve only y the trivial automorphism. The authors furthermore efine the istinguishing numer of G to e the minimum numer of colors in a istinguishing coloring. This parameter has een wiely stuie (for two nice examples, see [1] an [14]) an was quite recently extene to consier istinguishing colorings of a graph that are also proper vertex colorings. In [11], the authors efine the istinguishing chromatic numer of a graph G, enote χ D (G), to e the minimum numer of colors in a proper istinguishing coloring. In this paper we consier a variant of χ D (G), first introuce in [17]. Let C(v; χ) enote the multiset of colors assigne y a coloring χ to the neighors of vertex v. A proper coloring χ of G is irregular if χ(u) = χ(v) implies that C(u; χ) C(v; χ). The minimum numer of colors neee to irregularly color G is calle the irregular chromatic numer of G. Clearly, any irregular coloring of a graph G is also a istinguishing coloring, an it is therefore clear that χ(g) χ D (G) χ irr (G). However, χ irr (G) may not provie a very goo upper oun on χ D (G). In fact, these parameters may e aritrarily far apart. Given an irregular -coloring χ of a graph G an a vertex v in G, C(v) will e a multiset of carinality (v) with elements chosen from the 1 colors istinct from χ(v). We recall here that the numer of -element multisets over an alphaet of 1 symols is ( ) ( + 2 an note that this implies there can e at most + 2 )

3 DEBRUIJN-LIKE SEQUENCES AND IRREGULAR COLORINGS 3 vertices of egree in G that are assigne the same color y χ. This oservation was mae in the following lemma from [17]. Lemma 1. Let G e a graph an let e a positive integer. If ( ) + 2 n (G) > then χ irr (G) + 1. Lemma 1 serves to unerscore the istinction etween etermining χ(g) an χ irr (G). For instance, the wiely herale Four Color Theorem [5, 16] states that any planar graph can e properly colore with at most four colors. However, as the average egree of any planar graph is strictly less than six, Lemma 1 implies that if {G n } n 1 is an infinite sequence of planar graphs where each G i has orer i, then lim n χ irr (G n ) =. As another example, χ(p n ) = χ(c 2n ) = 2 an χ(c 2n 1 ) = 3 for all n 2. However, Lemma 1 implies that the irregular chromatic numer of these graphs is unoune as n tens to infinity. These graphs also allow us to contrast χ irr an χ D, as in [11] it was shown that for all n, χ D (P n ) 3 an that χ D (C n ) 4. This is unoutely ue to the fact that in computing χ irr (G) we wish to istinguish the vertices of G locally, while χ D affors the flexiility of a more gloal perspective. This example also emonstrates that the ae conition of irregularity may complicate the coloring of some graphs for which etermining χ or χ D is quite easy. In this paper, we etermine exactly the irregular chromatic numer of an aritrary path or cycle. A similar question was consiere [6] in the context of vertex-istinguishing ege colorings, an the similarities an ifferences etween the approaches there an those utilize in this paper help to unerscore the ifferences etween these parameters. In the next section, we efine a cominatorial structure that allows us to relate irregular colorings of paths an cycles to the existence of certain types of ebruijn-lie sequences. 2. The Overlap Digraph D 3 (A) Given an alphaet A, efine D 3 (A) to e the igraph that has as its vertex set the collection of orere three element strings a 1 a 2 a 3 from A with the property that a 1 a 2 an a 2 a 3. In general we will refer to these three element strings, an hence the vertices of D 3 (A), as wors over the alphaet A. If u = u 1 u 2 u 3 an v = v 1 v 2 v 3 are vertices in V (D 3 (A)), then uv is in A(D 3 (A)) if an only if u 2 = v 1 an u 3 = v 2. Since D 3 (A 1 ) an D 3 (A 2 ) are isomorphic if an only if A 1 = A 2, we will let D 3 () enote the overlap igraph constructe from an aritrary - element alphaet. If A = {a 1,..., a n } we may enote D 3 (A) y D 3 (a 1,..., a n ). The igraph D 3 (A) is an example of an overlap igraph, an is in fact an inuce sugraph of the classical 3-eBruijn igraph [12]. Our proof techniques will repeately utilize the structure of D 3 (A) when A = 3, as epicte in Figure 1.

4 4 M. FERRARA, E. GETHNER, C. LEE, P. WALLIS ac cc c ca ca ac cac a aca ca ac aa Figure 1. D 3 (a,, c) The lin etween overlap igraphs of this type an irregular coloring of paths an cycles is the focus of the results in this paper Overlap Digraphs an Irregular Colorings. For any wor w = a i a j a, efine the conjugate of w to e the wor w = a a j a i. If w = w we then say that w is self-conjugate or is a palinrome. A (irecte) cycle C in D 3 (A) is conjugateavoiing if for all w V (C), w V (C) if an only if w is self-conjugate. Similarly, a (irecte) path P is conjugate-avoiing if for every vertex w of egree 2 in P, w is also a vertex of egree 2 in P if an only if w is self-conjugate. In much the same way that each -ebruijn sequence correspons to an eulerian circuit in the classical -ebruijn igraph over A [12], each conjugate avoiing cycle can e associate with a ebruijn-lie sequence. The consecutive 3-element sustrings of this sequence correspon to a collection of wors, istinct up to conjugation, in which no two letters appear consecutively. Let V (C n ) = {v 1,..., v n } where the vertices are liste in the orer they appear on the cycle, an let χ : V (C n ) A e an irregular coloring of C n. For each vertex v i the string w i = χ(v i 1 )χ(v i )χ(v i+1 ), where the inices are taen moulo n, is a vertex in D 3 (A). Since χ is an irregular coloring of C n, for any two istinct wors w i an w j we have that w i w j an that w i w j. Inee, if this were not the case, the vertices v i an v j woul e assigne the same color y χ an their neighors woul also e assigne the same multiset of colors y χ. This woul imply that v i an v j have the same color coe, contraicting the irregularity of χ. In fact, it is not ifficult to see that for each i, w i w i+1 is an arc in D 3 (A) an, moreover, the vertices w 1,...,w n in this orer form a cycle C. By the previous iscussion, the irregularity of χ implies that C is conjugateavoiing. An ientical analysis yiels that if C is a conjugate-avoiing cycle of length n in D 3 (), then this cycle gives an irregular coloring of C n using at most colors. The following lemma summarizes these oservations an an example is given in Figure 2. Lemma 2. Let n an 3 e positive integers. Then χ irr (C n ) if an only if D 3 () contains a conjugate-avoiing cycle of length n. The prolem of etermining the irregular chromatic numer of P n is similar to that of etermining the irregular chromatic numer of C n, ut is somewhat complicate y the presence of the two en-vertices. Again, let V (P n ) = {v 1...., v n } where the vertices are inexe y the orer they appear on the path an let

5 DEBRUIJN-LIKE SEQUENCES AND IRREGULAR COLORINGS 5 a a ac c a c c Figure 2. An irregular coloring of C 6 an the associate conjugate-avoiing cycle in D 3 (4) χ : V (P n ) A e an irregular coloring of P n. For each vertex v i where 2 i n 1 (the vertices of egree 2 in P n ), the string w i = χ(v i 1 )χ(v i )χ(v i+1 ) represents a vertex in D 3 (A). Since χ is an irregular coloring, each of these w i must e istinct an for each w i an w j we must have that w i w j. For each i with 1 i n 1 we have that w i w i+1 is an arc in D 3 (A) which implies that the vertices w i, in orer, form a conjugate-avoiing path of length n 2 in D 3 (A). Prior to iscussing those conjugate-avoiing paths in D 3 (A) that can e use to generate irregular colorings of paths, we nee to consier the color coes of v 1 an v n, which are the vertices of egree one in P n. Since χ is an irregular coloring, either χ(v 1 ) χ(v n ) or χ(v 2 ) χ(v n 1 ) (or possily oth). We efine a conjugate-avoiing path from w 1 = a 1 a 2 a 3 to w = a a +1 a +2 in D 2 (A) to e ounary istinguishe if either a 1 a +2 or a 2 a +1. This aitional conition allows us to characterize those conjugate-avoiing paths in D 3 (A) that give rise to irregular colorings of P n. Lemma 3. Let 3 an n e positive integers. Then χ irr (P n ) if an only if D 3 () contains a ounary istinguishe conjugate-avoiing path on n 2 vertices. In the following sections, we will construct conjugate-avoiing paths an cycles in D 3 () an use them to etermine the irregular chromatic numer of paths an cycles. 3. Irregular Colorings of Cycles The irregular chromatic numer of paths an cycles was iscusse in oth [15] an [17]. For the remainer of this paper, we let n enote the quantity (. The main result of this section is as follows. Theorem 1. Let 4 an n n n e integers. Then { if n n 1 χ irr (C n ) = + 1 if n = n 1. We note that Theorem 1 verifies a conjecture given in [15]. Proof. Let an n, where n n n, e as given in the statement of the theorem. In light of Lemma 2 we note that via an elementary counting argument, one can see that the maximum size of a conjugate-free suset of V (D 3 ( 1)), an hence the maximum possile length of a conjugate-avoiing cycle in D 3 ( 1),

6 6 M. FERRARA, E. GETHNER, C. LEE, P. WALLIS is n 1. Consequently, as n > n 1 it must e that χ irr (C n ). In line with Lemma 2, the proof procees y constructing conjugate-avoiing cycles of lengths 3,...,n 2 an n in D 3 (). Let A = {a 1,..., a } e a -element alphaet. Each of the 3-element susets of A can e associate with a unique sugraph of D 3 (A) that is isomorphic to D 3 (. The sugraphs associate with the 3-element susets S i an S j intersect if an only if S i S j = 2. With this oservation in min, we efine the auxiliary graph of D 3 (), enote G aux (), to e the graph whose vertices are 3-element susets of A such that two vertices are ajacent if an only if their corresponing sets share exactly two elements. The structure of G aux () mimics the structure of D 3 (), where the vertices of G aux () represent istinct copies of D 3 ( in D 3 (). For a vertex v in V (G aux ()), we efine the preimage of v to e the copy of D 3 ( in D 3 () that is associate with v. Similarly, for any sugraph H of G aux (), we efine the preimage of H to e the sugraph of D 3 () spanne y the preimages of the elements of V (H). It will e useful to more closely examine the structure of D 3 (, with Figure 1 serving as a helpful reference. The vertices of D = D 3 (a,, c) can e partitione into the two irecte triangles ac, ca, ca an ca, ac, ac, henceforth referre to as the central triangles of D, along with three pairs of palinromes, each etermine y a 2-element suset of {a,, c}. The next claim allows us to use the structure of G aux () to construct cycles in D 3 () using only the vertices in central triangles in some or all of the copies of D 3 (. For convenience, we exten the efinition of a central triangle as follows. If a conjugate-avoiing cycle C in D 3 () has the property that V (C) = V (τ 1 ) V (τ 2 ) V (τ t ), where each τ i is a central triangle in some copy of D 3 (, then we will say that C is central. Clearly, if C is central, then V (C) = 3t for some integer t. Also, each of the triangles τ i must e selecte from istinct copies of D 3 ( in D 3 (), as C is conjugate-avoiing. For clarity, we point out that while V (C) is the isjoint union of the sets V (τ i ), if V (C) 3 it will not e the case that E(C) is the isjoint union of the sets E(τ i ). Claim 1. Let 3 an t e integers such that 1 t (. Then there is a conjugate-avoiing cycle C of length 3t in D 3 () such that C is central. Proof. Let T e any spanning tree of G aux (), an efine a sequence of trees T 1 T ( such that each T i has orer i an T ( = T. We will show that there is a central conjugate-avoiing cycle of length 3t in the preimage of T t for each 1 t (. We procee y inuction on t, an note that the claim hols when t = 1. Thus, let C e a central conjugate-avoiing cycle in the preimage of T t for t 1 an let v = V (T t+1 ) \ V (T t ). Furthermore, let u e a neighor of v in T t+1. Assume that the preimage of v is D 3 (a 1, a 2, a 3 ) an that the preimage of u is, without loss of generality, D 3 (a 1, a 2, x). Since C is central, the vertices of one of the central triangles of D 3 (a 1, a 2, x) are also vertices in V (C). Without loss of

7 DEBRUIJN-LIKE SEQUENCES AND IRREGULAR COLORINGS 7 generality, assume that the vertices a 1 a 2 x, a 2 xa 1 an xa 1 a 2 are in V (C). Then there is some y (possily equal to x) such that the arc from xa 1 a 2 to a 1 a 2 y is in C. We will augment C y removing the arc from xa 1 a 2 to a 1 a 2 y an aing the arcs (xa 1 a 2, a 1 a 2 a 3 ), (a 1 a 2 a 3, a 2 a 3 a 1 ), (a 2 a 3 a 1, a 3 a 1 a 2 ) an (a 3 a 1 a 2, a 1 a 2 y). As C was central an conjugate-avoiing, this results in a central conjugateavoiing cycle of length 3t + 3 in the preimage of T t+1. To emonstrate the augmentation process escrie in Claim 1, Figure 3 shows a central triangle in D 3 (a,, c) extene to a central 6-cycle in the sugraph of D 3 (4) inuce y D 3 (a,, c) an D 3 (a,, ). ca ca a ca ca a ac ac a Figure 3. Constructing a central cycle of length six from a central cycle of length three We now show that it is possile to increase the lengths of central cycles, an many other types of cycles, through the aition of pairs of self-conjugate vertices. Claim 2. Let A e an alphaet an suppose a 1, a 2 an x are istinct elements in A. If C is a conjugate-avoiing cycle in D 3 (A) that contains the vertex a 1 a 2 x ut oes not contain the palinromes a 1 a 2 a 1 an a 2 a 1 a 2, then there is a conjugate-avoiing cycle C in D 3 (A) such that V (C ) = V (C) {a 1 a 2 a 1, a 2 a 1 a 2 }. Proof. Let C e as given, an let y A e such that ya 1 a 2 is the preecessor of a 1 a 2 x in C. We construct C y removing the arc from ya 1 a 2 to a 1 a 2 x from C an aing the arcs (ya 1 a 2, a 1 a 2 a 1 ), (a 1 a 2 a 1, a 2 a 1 a 2 ) an (a 2 a 1 a 2, a 1 a 2 x). Claim 2 allows us to construct conjugate-avoiing cycles of length 3t, 3t+2 an 3t + 4 for 1 t ( in D3 () y aing either one or two pairs of palinromes to the conjugate-avoiing cycle of length 3t assure y Claim 1. It is not ifficult to see that this implies that D 3 () contains conjugate-avoiing cycles of length l for any l etween 3 an 3 ( except for l = 4, 6 or 8. Inspection of Figure 1 yiels that D 3 ( oes not contain conjugate-avoiing cycles of length 4, 6 or 8, while the irregular colorings given in Figure 4 emonstrate that cycles of this length o appear in D 3 () for 4. We now wish to emonstrate the existence of cycles of length 3 (,..., n 2 an n in D 3 (). Let C an C enote central conjugate-avoiing cycles in D 3 () of length 3 ( ( an 3 3 respectively. Since C is a central cycle of maximum length, we note that C contains the vertices of a central triangle from each copy of D 3 ( containe in D 3 (). Furthermore, note that C contains the vertices of a central triangle from all ut exactly one of the copies of D 3 ( containe in D 3 (). This implies that oth C an, since 4, C contain vertices of the form a 1 a 2 x for every choice of a 1 an a 2 in an appropriate -element alphaet A.

8 8 M. FERRARA, E. GETHNER, C. LEE, P. WALLIS a a a a c c c c Figure 4. Irregular 4-colorings of C 4, C 6 an C 8 We may therefore procee to augment oth C an C using Claim 2. By aing each of the ( pairs of palinromes in D3 () to C, it is possile to create conjugateavoiing cycles of length 3 ( ( +2, 3 ( +4...,3 ( +2 = n in D 3 (). Similarly, y aing these pairs of palinromes to C, it is possile to create conjugateavoiing cycles of length 3 ( ( 1, 3 ( + 1,...,3 ( = n 3. It is not ifficult to see that this emonstrates the existence of conjugate-avoiing cycles of lengths 3 (,..., n 2 an n in D 3 (). Together with the shorter cycles constructe aove, this emonstrates that D 3 () contains conjugate-avoiing cycles of length 3,...,n 2 an n. We complete the proof y noting that Theorem 3.3 from [17] shows that for any 3, χ irr (C n 1) +1. This implies that D 3 () contains no conjugate-avoiing cycle of length n 1. We have shown, however, that there is a conjugate-avoiing cycle of length n 1 in D 3 ( + 1) an hence that χ irr (C n 1) = + 1. For completeness, we state the following. Corollary 2. Let n 3 an 3 e integers, an let n = (. Then 3 n = 3, 5, 7, 9 4 n = 4, 6, 8 χ irr (C n ) = 4 an either n n n 2 or n = n an n = n Irregular Colorings of Paths In this section, we are intereste in etermining χ irr (P n ) for all n 2. It is not ifficult to see that χ irr (P 2 ) = χ irr (P 4 ) = 2 an that χ irr (P 3 ) = 3. For n 5 we will utilize Lemma 3 an emonstrate the existence of ounary istinguishe conjugate-avoiing paths of orer n 2 in D 3 () for an appropriate choice of. The main result of this section is as follows, an serves to verify a conjecture given in [15]. Theorem 3. Let 3 an n e integers such that n n + 2 e integers. Then χ irr (P n ) =. Proof. For 5 n 11, Theorem 3 can e easily verifie y inspection of Figure 1, which yiels ounary istinguishe conjugate-avoiing paths of length 3,...,9 in D 3 (. For n 12, an hence 4, Theorem 3 follows almost immeiately

9 DEBRUIJN-LIKE SEQUENCES AND IRREGULAR COLORINGS 9 from Theorem 1 an the following lemma, which appears as Proposition 4.3 in [15]. We give an alternate proof here that utilizes the oservations mae aove relating χ irr (P n ) to the structure of D 3 (). Lemma 4. For each integer n 5, χ irr (P n ) χ irr (C n 2 ). Proof. Let n e as given, an let χ irr (C n 2 ) = t. Let C e the conjugate-avoiing cycle in D 3 (t) associate with some irregular t-coloring of C n 2 an let a 1 a 2 a 3 an a 2 a 3 x e consecutive vertices on C. Let P e the path in D 3 (t) otaine y eleting the arc (a 1 a 2 a 3, a 2 a 3 x) from C. As C is conjugate-avoiing, so too is P, an since a 2 a 3, we conclue that P is ounary istinguishe. Thus P is a ounary istinguishe conjugate-avoiing path of orer n 2 in D 3 (t), implying that χ irr (P n ) t. It was shown in Theorem 1 that when 4, D 3 () contains conjugate-avoiing cycles of length 3,...,n 2 an n. This, along with Lemma 4, implies that if 5 n n or if n = n + 2 then χ irr (P n ). It remains to show that if n = n + 1 then χ irr (P n ). Inee, let C e a conjugate-avoiing cycle of length n in D 3 () an let a 1 a 2 a 3, a 2 a 3 a 4 an a 3 a 4 a 5 e three consecutive vertices on C. Removing a 2 a 3 a 4 from C will result in a conjugateavoiing path P an P will e ounary istinguishe provie a 2 a 4. That is, the removal of any non-palinrome from C will result in a ounary istinguishe conjugate-avoiing path of orer n 1 in D 3 (), implying that χ irr (P n +1). In particular, the preceing analysis yiels that if n n n + 2 then χ irr (P n ). Lemma 1 (with r = implies that for these values of n, χ irr (P n ). The result follows. 5. Conclusion It woul e of interest to exten the class of conjugate-free ebruijn-lie sequences to inclue those cyclic sequences whose t-element sustrings are istinct up to conjugation for some t > 3. At this time, we o not have a particular application of these sequences in min, ut it is reasonale to thin that one may e foun. The notion of an irregular vertex coloring of a graph is also relatively new, an we elieve that the area remains ripe for fruitful investigations in the future. A natural, an perhaps approachale, extension of the results in this paper woul e to etermine the irregular chromatic numer of an aritrary graph G with (G) 2; that is, G is an aritrary union of paths an cycles. Let G e the graph whose components are the paths P i1,...,p i an the cycles C j1,...,c jt where the inices are all at least three an are not necessarily istinct. In much the same manner as Lemma 2 an Lemma 3, an irregular -coloring of G correspons to a sugraph H of D 3 () compose of conjugate-avoiing cycles an ounary istinguishe conjugate-avoiing paths with the ae property that if w an w are in H then w = w. As implie y Lemma 1, if such a -coloring of G were to exist then n 2 (G) (. We conjecture that this is nearly optimal.

10 10 M. FERRARA, E. GETHNER, C. LEE, P. WALLIS Conjecture 1. Let G e a graph with (G) 2 an let e the unique integer such that ( ) ( ) 1 ( 1) + 1 n 2 (G). 2 2 Then χ irr (G) + 1. Also, while this paper was uner review, we receive notice that Theorems 1 an 3 were otaine inepenently in [3]. While [3] i not utilize structure ebruijn sequences, the intereste reaer may wish examine the approach there, as it provies an interesting contrast to the techniques employe in this paper. Acnowlegement: The authors woul lie to than the referees for their thoughtful comments, which improve oth the exposition an clarity of this paper. References [1] M. Alertson, D. Boutin, Distinguishing geometric graphs, J. Graph Theory 53 (2006), no. 2, [2] M. Alertson, K. Collins, Symmetry reaing in graphs. Electron. J. Comin. 3 (1996) R18. [3] M. Anerson, C. Barrientos, R. Brigham, J. Carrington, M. Kronman, R. Vitray an J. Yellen, The Irregular Chromatic Numer of Some Graph Classes, to appear in Bull. Inst. Comin. Appl. [4] J. Anerson, K. Fox, G. Nilo, A fast algorithm for the construction of universal footprinting templates in DNA. J. Math. Biol. 52 (2006), no. 3, [5] K. Appel, W. Haen. Every planar map is four colorale. Bull. Amer. Math. Soc. 82 (1976), no. 5, [6] P. Balister, B. Bollos, R.Schelp, Vertex istinguishing colorings of graphs with (G) = 2. Discrete Math. 252 (200, no. 1-3, [7] A. Burris, R. Schelp. Vertex-istinguishing proper ege colorings. J. Graph Theory 26 (1997) no 2, [8] G. Chartran, H. Escuaro, F. Oamoto, P. Zhang. Detectale colorings of graphs, Util. Math. To appear. [9] G. Chartran an L. Lesnia. Graphs an Digraphs, Fourth Eition. Chapman & Hall/ CRC. (2005). [10] G. Chartran, M. S. Jacoson, J. Lehel, O. Oellerman, S. Ruiz an F. Saa, Irregular Networs, Congr. Num., 64 (1988) [11] K. Collins, A. Tren. The istinguishing chromatic numer. Electron. J. Comin. 13 (2006), no. 1, R16. [12] N. G. ebruijn. A cominatorial prolem, Koninliije Neerlanse Aaemie v. Wetenschappen, 49 (1946) [13] F Harary, R.A. Melter. On the metric imension of a graph. ARS Comin. 2 (1976) [14] W. Imrich, S. Klaˇvzar, Distinguishing Cartesian powers of graphs, J. Graph Theory 53 (2006), no. 3, [15] F. Oamoto, M. Racliffe, P. Zhang. On the Irregular Chromatic Numer of a Graph. Congr. Num, 181 (2006) [16] N. Roertson, D. Saners, P. Seymour, R. Thomas. A new proof of the four-colour theorem. Electron. Res. Announc. Amer. Math. Soc. 2 (1996), no. 1, [17] M. Racliffe, P. Zhang. Irregular Colorings of Graphs, Bull. Inst. Comin. Appl., 49 (2007) [18] P. Slater. Leaves of Trees. Congr. Num, 14 (1975)