Total tessellation cover and quantum walk
Alexandre Abreu, Luís Cunha, Celina de Figueiredo, Franklin Marquezino, Daniel Posner, Renato Portugal
TTotal tessellation cover and quantum walk (cid:63)
A. Abreu , L. Cunha , C. de Figueiredo F. Marquezino , D. Posner , and R. Portugal Universidade Federal do Rio de Janeiro – { santiago, lfignacio, celina,franklin, posner } @cos.ufrj.br Laborat´orio Nacional de Computa¸c˜ao Cient´ıfica – [email protected]
Abstract.
We propose the total staggered quantum walk model and thetotal tessellation cover of a graph. This model uses the concept of totaltessellation cover to describe the motion of the walker who is allowed tohop both to vertices and edges of the graph, in contrast with previousmodels in which the walker hops either to vertices or edges. We establishbounds on T t ( G ), which is the smallest number of tessellations requiredin a total tessellation cover of G . We highlight two of these lower bounds T t ( G ) ≥ ω ( G ) and T t ( G ) ≥ is ( G ) + 1, where ω ( G ) is the size of amaximum clique and is ( G ) is the number of edges of a maximum inducedstar subgraph. Using these bounds, we define the good total tessellablegraphs with either T t ( G ) = ω ( G ) or T t ( G ) = is ( G ) + 1. The k -totaltessellability problem aims to decide whether a given graph G has T t ( G ) ≤ k . We show that k -total tessellability is in P for good totaltessellable graphs. We establish the N P -completeness of the followingproblems when restricted to the following classes: ( is ( G ) + 1 )-totaltessellability for graphs with ω ( G ) = 2; ω ( G ) -total tessellability for graphs G with is ( G ) + 1 = 3; k -total tessellability for graphs G with max { ω ( G ) , is ( G ) + 1 } far from k ; and 4 -total tessellability for graphs G with ω ( G ) = is ( G ) + 1 = 4. As a consequence, we establishhardness results for bipartite graphs, line graphs of triangle-free graphs,universal graphs, planar graphs, and (2 , Keywords:
Graph tessellation, Quantum walk, Graph coloring, Com-putational complexity. A tessellation of a graph G = ( V, E ) is a partition of V into vertex disjoint cliquescalled tiles . A k -tessellation cover of G is a set of k tessellations that covers E .The tessellation cover number T ( G ) of a graph G is the size of a minimum tes-sellation cover. The k -tessellability problem aims to decide whether a givengraph G has T ( G ) ≤ k . The concept of tessellations on graphs was introducedin [1]. See [2] for basic definitions and notations in graph theory. (cid:63) This work was partially supported by the Brazilian agencies CAPES, CNPq andFAPERJ. a r X i v : . [ c s . D M ] F e b A. Abreu et al.
Definition 1
Let G = ( V, E ) be a graph and Σ a non-empty label set. A totaltessellation cover comprises a proper vertex coloring and a tessellation cover of G both with labels in Σ such that, for any vertex v ∈ V , there is no edge e ∈ E incident to v so that e belongs to a tessellation with label equal to the color of v .An alternative way to characterize a tessellation is by describing the edgesthat belong to the tessellation. A k -tessellation cover of G = ( V, E ) is a function h that assigns to each edge of E a nonempty subset in P ( Σ ), where Σ = { , . . . , k } ,such that the set of edges having the same label corresponds to a tessellation,i.e., induces a partition of V into cliques. A k -total tessellation cover of a graph G simultaneously assigns labels in Σ to V as a proper vertex coloring f andlabels in P ( Σ ) \ ∅ to E as a tessellation cover with function h , such that each uv ∈ E satisfies f ( u ) (cid:54)∈ h ( uv ) and f ( v ) (cid:54)∈ h ( uv ). Definition 2
The total tessellation cover number T t ( G ) of a graph G is theminimum size of the set of labels Σ for which G has a total tessellation cover.The k -total tessellability problem aims to decide whether a given graph G has T t ( G ) ≤ k . Motivation.
The quantum computation paradigm has gained popularity dueto the recent advances in the physical implementation and in the development ofquantum algorithms. There is an important concept, known as quantum walk,which is the mathematical modeling of a walk of a particle on a graph. Thisconcept provides a powerful tool in the development of quantum algorithms [3].Indeed, in the last decades the interest in quantum walks has grown considerablysince quantum algorithms that outperform their classical counterparts employquantum walks [4, 5]. In 2016, Portugal et al. proposed the staggered quantumwalk model [1], which is more general than the previous quantum walk models [6]by containing the Szegedy model [7] and part of the flip-flop coined model [3].The staggered quantum walk employs the concept of graph tessellation cover toobtain local unitary matrices such that their product results in the evolutionoperator for the quantum walk. There is a recipe to obtain a local unitary ma-trix from a tessellation. The staggered model requires at least two tessellations(corresponding to 2-tessellable graphs). In a tessellation, each clique establishesa neighborhood around which the walker can move under the action of the asso-ciated local unitary matrix. To define the evolution operator, one has to checkwhether the set of tessellations contains the whole edge set of the graph, sincean uncovered edge would play no role in a quantum walk [1].
Related works.
Abreu et al. [8, 9] proved that χ (cid:48) ( G ) and χ ( K ( G )) are up-per bounds for T ( G ), where K ( G ) is the clique graph of G . They also provedthe hardness of k -tessellability for planar graphs, (2 , , -tessellability is solved in linear time. Since T ( G ) = χ (cid:48) ( G ) for triangle-free graphs, k -tessellability is hard for this graphclass [10]. Posner et al. [11] showed that k -tessellalbility is N P -complete for otal tessellation cover and quantum walk line graphs of triangle-free graphs. Abreu et al. [12] proved that is ( G ) is a lowerbound for T ( G ), where is ( G ) is the number of edges in a maximum inducedstar of a graph G . They prove the hardness of k -tessellability for universalgraphs and the hardness of good tessellable recognition , which aims todecide whether G is good tessellable , i.e., T ( G ) = is ( G ). The concept of mini-mum tessellation cover was independently proposed as equivalence dimension inDuchet [13], and the relation between the two concepts was described in [12]. Contributions.
This work is presented in the following sections. Section 2contains a study on the bounds of the value of T t ( G ). Such bounds describe notonly the number of operators required for a total staggered quantum walk model,but they also provide tools to analyse the computational complexity of k -totaltessellability , which is done in Section 3. Since T t ( G ) = χ t ( G ) for triangle-free graphs, the problem is hard even when restricted to bipartite graphs [14]. Weshow that k -total tessellability is in P for good total tessellable graphs,and as a by product k -tessellability is in P for good tessellable graphs.On the other hand, we show hardness results for the k -total tessellability problem for line graphs of triangle-free graphs, universal graphs, planar graphs,and (2 , good total tessellabilityrecognition problem is N P -complete. Note that there are few results aboutthe hardness of total colorability . Section 4 describes the total staggeredquantum walk model, which drives a walker to hop both to vertices and edges.It also contains a description of the simulation of the total staggered quantumwalk on a graph G in terms of a staggered quantum walk on the total graphof G . In Section 5, Table 1 presents the behavior analysis of the computationalcomplexity related to the following parameters: χ (cid:48) ( G ) , χ t ( G ) , T ( G ), and T t ( G ). T t ( G ) Since a total coloring of a graph G induces a total tessellation cover, T t ( G ) ≤ χ t ( G ) . (1)Particularly, for triangle-free graphs T t ( G ) = χ t ( G ) because the set of edges ineach tessellation of any total tessellation cover is a matching. Hence, ( ∆ +1 )-total tessellability is hard even when restricted to regular bipartitegraphs [14]. Furthermore, by definition,max { χ ( G ) , T ( G ) } ≤ T t ( G ) ≤ χ ( G ) + T ( G ) . (2)Note that the lower bound of Eq. (2) implies that T t ( G ) ≥ ω ( G ). Lemma 1. If χ ( G ) ≥ T ( G ) , then T t ( G ) = χ ( G ) . Proof.
Let f be a proper vertex coloring and C = {T , T , . . . , T T ( G ) } be a T ( G )-tessellation cover for G . We define C (cid:48) a tessellation cover for G with 3 T ( G ) labels A. Abreu et al. such that C (cid:48) is compatible with f as follows. Each tessellation T (cid:48) i , ≤ i ≤ T ( G ),of C (cid:48) is associated with a color i . Since χ ( G ) ≥ T ( G ) there are enough colors.The edges of tessellations T (cid:48) j − , T (cid:48) j − , and T (cid:48) j are given by the edges of thetessellation T j , ≤ j ≤ T ( G ), such that T (cid:48) j − (resp. T (cid:48) j − , T (cid:48) j ) consists of theedges of T j that do not have an endpoint with color 3 j − j − , j ). (cid:117)(cid:116) Using an argument similar to the one in the proof of Lemma 1, we can rewritethe upper bound of Eq. (2) as follows T t ( G ) ≤ max { χ ( G ) , T ( G ) + (cid:100) χ ( G ) / (cid:101)} . (3)Eq.(3) says that χ ( G ) ≥ T ( G ) implies T t ( G ) = χ ( G ), or χ ( G ) ≤ T ( G ) im-plies T ( G ) ≤ T t ( G ) ≤ T ( G ). In case χ ( G ) = 3, Eq. (3) implies that T ( G ) ≤ T t ( G ) ≤ T ( G ) + 2. An example of a graph G for which T t ( G ) = 3 T ( G ) − T t ( G ) > χ ( G ) has V ( G ) = { v , v , v , v } ∪ { u , u , u , u } ∪ { w , w , w , w } ,where { v , v , v , v } and { u , u , u , u } are maximal cliques and { v i , u i , w i } are triangles for 1 ≤ i ≤
4. In this case T t ( G ) = 5, χ ( G ) = 4 and T ( G ) = 2.Note that T t ( G ) = χ ( G ) + T ( G ) requires that χ ( G ) ≤
2, i.e., G is bipar-tite, which implies T t ( G ) = χ t ( G ) and T t ( G ) may assume only two values: T t ( G ) = χ ( G ) + T ( G ) = ∆ ( G ) + 2 or T t ( G ) = χ ( G ) + T ( G ) − ∆ ( G ) + 1. Lemma 2. T t ( G ) ≥ max v ∈ V ( G ) { χ ( G c [ N ( v )]) } +1 ≥ max v ∈ V ( G ) { ω ( G c [ N ( v )]) } +1 = is ( G )+1 . Proof.
Consider a total tessellation cover of a graph G , a vertex v of G , and G c [ N ( v )], which is the complement graph of the graph induced by the neighbor-hood of v . In any tessellation, the endpoints of the edges that are incident to v and belong to the tessellation induce a clique, hence the vertices of this clique area stable set in G c [ N ( v )]. Therefore, the tessellations with edges incident to a ver-tex v induce a vertex coloring of G c [ N ( v )], and the number of these tessellationsis at least χ ( G c [ N ( v )]). Moreover, these tessellations have labels that are differ-ent from the color of vertex v . Therefore, T t ( G ) ≥ χ ( G c [ N ( v )]) + 1. Note that is ( G [ N [ v ]]) = α ( G [ N ( v )]) = ω ( G c [ N ( v )]) and is ( G ) = max v ∈ V ( G ) is ( G [ N [ v ]]). (cid:117)(cid:116) Graphs with T t ( G ) = T ( G ) = k have no induced subgraph K ,k because T t ( G ) ≥ is ( G ) + 1 ≥ k + 1. Moreover, there is no tile of size k in any tessellationof a total tessellation cover. If T t ( G ) = T ( G ) = 3, then G is K , -free and there isno clique of size three in any tessellation. Therefore, the total tessellation cover of G induces a total coloring of G , and the only graphs for which T t ( G ) = T ( G ) = 3are the odd cycles with n vertices such that n ≡ T ( G ) = ∆ ( G ) and T t ( G ) > T ( G ). For triangle-free graphs, T t ( G ) = T ( G ) if χ (cid:48) ( G ) = χ t ( G ) = ∆ + 1. It follows that deciding whether T t ( G ) = T ( G ) = ∆ ( G ) + 1 is N P -complete from the proof that ( ∆ + 1 )-total colorability is N P -complete for triangle-free snarks [15], which are graphs with χ (cid:48) ( G ) = ∆ + 1. Since the concept of good tessellable graphs introduced in [12] has provided keeninsights into the hardness of finding minimum-sized tessellation covers, we define otal tessellation cover and quantum walk the concept of good total tessellable graphs in order to further explore hardnessresults related to total tessellation covers. In the quantum computation context,we are interested in graph classes which use as few color labels as possible becausethe number of operators is as low as possible. In this case, T t ( G ) must be closeto the lower bounds. Definition 3
A graph G is good total tessellable if either T t ( G ) = ω ( G ) or T t ( G ) = is ( G ) + 1. We say that G is Type I (resp.
Type II ) if T t ( G ) = ω ( G )(resp. T t ( G ) = is ( G ) + 1).Now we show that k -total tessellability is in P if we know beforehandthat the graph is either good total tessellable Type I or Type II.The Lov´asz number ϑ ( G ) is a real number such that ω ( G c ) ≤ ϑ ( G ) ≤ χ ( G c ) [16]. We denote ψ ( G ) the integer nearest to ϑ ( G ). The value of ψ ( G )can be be determined in polynomial time [16].For Type I graphs, T t ( G ) = ω ( G ). Since Eq. (2) implies that ω ( G ) ≤ χ ( G ) ≤ T t ( G ), we have ω ( G ) = χ ( G ) = T t ( G ) = ψ ( G c ).For Type II graphs, T t ( G ) = is ( G )+1. For any vertex v ∈ V ( G ), ω ( G c [ N ( v )]) ≤ ψ ( G [ N ( v )]) ≤ χ ( G c [ N ( v )]), and by Lemma 2, T t ( G ) ≥ ψ ( G [ N ( v )]) + 1. Since T t ( G ) = is ( G ) + 1, by Lemma 2 there is a vertex u ∈ V ( G ) such that T t ( G ) = ω ( G c [ N ( u )]) + 1. In this case, ω ( G c [ N ( u )]) + 1 = χ ( G c [ N ( u )]) + 1, and we deter-mine ω ( G c [ N ( u )]) using ψ ( G c [ N ( u )]). Therefore, T t ( G ) = max v ∈ V ( G ) { ψ ( G [ N ( v )]) } +1.The same method used to determine T t ( G ) for Type II graphs can be ap-plied for good tessellable graphs in order to determine T ( G ), where T ( G ) =max v ∈ V ( G ) { ψ ( G [ N ( v )]) } . Hardness results.
As presented in Section 2, ( ∆ + 1)- total tessellability is N P -complete for bipartite graphs, which have is ( G )+1 = ∆ +1 and ω ( G ) = 2.Now, we show that k -total tessellability is N P -complete for the followingcases: line graph of triangle-free graphs with k = ω ( G ) ≥ is ( G ) + 1 = 3;universal graphs with k very far apart from both is ( G ) + 1 and ω ( G ); planargraphs with k = 4 = ω ( G ) = is ( G ) + 1; and (2 , k = is ( G ) + 1 = ω ( G ) + 3. Line graph of triangle-free graphs.
Machado et al. [17] proved that k - edgecolorability is N P -complete for 3-colorable k -regular triangle-free graphs if k ≥
3. The key idea of the proof of Theorem 1 is to verify that T t ( L ( G )) = χ (cid:48) ( G )when k ≥
9. The edges incident to any vertex v of graph G correspond to a cliqueof L ( G ), whose size is the degree of v . If two vertices of G are non-adjacent, thenthe corresponding cliques in L ( G ) share no vertices. Hence, we cover the edges ofthe cliques of L ( G ) incident to the vertices of each of the three color class of the3-coloring of G with a tessellation related to the color class because these cliquesshare no vertices. Therefore, since T ( L ( G )) = 3 and χ ( L ( G )) ≥ ≥ T ( G ), byLemma 1, T t ( L ( G )) = χ ( L ( G )) = χ (cid:48) ( G ). Note that in this case k = ω ( L ( G ))and is ( L ( G )) + 1 = 3. A. Abreu et al.
Theorem 1. k -total tessellability is N P -complete for line graphs L ( G ) of -colorable k -regular triangle-free graphs G for any k ≥ . Universal graphs.
Abreu et al. [12] reduced q - colorability to k - tessellability for universal graphs. We present a similar argument to establish the N P -completeness of k -total tessellability for universal graphs. Let G be aninstance of q - colorability . The key idea of the proof of Theorem 2 is to addto G c a universal vertex u and 2 | V ( G ) | pendant vertices adjacent to u , whichdefines the graph [2 | V ( G ) | , G c ] of G c . Now, the total tessellation cover number ofthe constructed graph is given by 2 | V ( G ) | + χ ( G ) + 1, using labels 1 , . . . , χ ( G ) tocover the edges incident to u that belong to the subgraph induced by V ( G c ∪{ u } ),labels χ ( G ) + 1 , . . . , χ ( G ) + 2 | V ( G ) | to cover the edges incident to the pendantvertices and labels χ ( G ) + 1 , . . . , χ ( G ) + | V ( G ) | are enough to cover the edgesof G c ; assign to u the color 2 | V ( G ) | + χ ( G ) + 1, to the pendant vertices color1, and to the remaining vertices colors χ ( G ) + | V ( G ) | + 1 , . . . , χ ( G ) + 2 | V ( G ) | .The minimality follows from Lemma 2. Therefore, T t ([2 | V ( G ) | , G c ]) = 2 | V ( G ) | + χ ( G ) + 1.Note that is ( C ∨ { u } ) = 2, T t ( C ∨ { u } ) = 4, and any minimum totaltessellation cover of C ∨ { u } has at least three labels assigned to the edgesincident to u and a fourth label assigned to u . Thus, T t ([2 | V ( G ) | , G c ∪ C ]) = T t ([2 | V ( G ) | , G c ]) + 3; is ([2 | V ( G ) | , G c ∪ C ]) = is ([2 | V ( G ) | , G c ]) + 2; and ω ([2 | V ( G ) | , G c ∪ C ]) = ω ([2 | V ( G ) | , G c ]). Therefore, each addition of a C in-creases the gap between the total tessellation cover number and both the sizesof a maximum induced star and a maximum clique. As long as the number ofthe C ’s is polynomially bounded by the size of G , k -total tessellability is N P -complete even if k is far apart from is ( G ) and ω ( G ). Theorem 2. k -total tessellability is N P -complete for universal graphs.
Planar graphs.
We show that 4 -total tessellability is N P -complete whenrestricted to planar graphs G with is ( G ) + 1 = ω ( G ) = 4. We present a poly-nomial transformation from 3 -colorability when restricted to planar graphswith maximum degree four [18] to 4 -total tessellability for planar graphs.Let G be an instance of such coloring problem. G (cid:48) = G ∨ { u } has a 4-coloringif and only if the planar graph G has a 3-coloring. We define three gadgets asdepicted in Fig. 1. The edges of the external triangles of the Duplicator Gadget are tiles of size three in a same tessellation. The edges of the external triangles ofthe
NotEqual Gadget are tiles of size three in different tessellations. The
ShifterGadget forces triangles T and T to be tiles on a tessellation a , and triangles T and T to be tiles on a tessellation b different from a .Each vertex v of G (cid:48) is associated with a Duplicator Gadget such that thenumber of external triangles of the Duplicator Gadget of v is equal to d G (cid:48) ( v ).If two vertices of G (cid:48) are adjacent, we connect one external triangle of eachDuplicator Gadget with a NotEqual Gadget. Thus, in a 4-total tessellation coverof the obtained graph H , the labels of the external triangles of the Duplicator otal tessellation cover and quantum walk >Duplicator Gadget NotEqual Gadget Shifter Gadget T T T T Fig. 1.
Duplicator Gadget, NotEqual Gadget, and Shifter Gadget.
Gadget associated with a vertex v are equal to the color of v in a 4-coloringof G (cid:48) . Now, we transform H into a planar graph H (cid:48) by replacing each crossingtriangles of H by a Shifter Gadget. Therefore, the planar graph H (cid:48) has a 4-total tessellation cover if and only if G has a 3-coloring. Note that in this case k = ω ( G ) = 4 = is ( G ) + 1. Theorem 3. -total tessellability is N P -complete for planar graphs. (2 , A graph G is (2 ,
1) if its vertex set can be parti-tioned into two stable sets and one clique. Since 3 -edge colorability is N P -complete for 3-regular graphs [17], 3 -vertex colorability for 4-regularline graphs is also
N P -complete. Let G be a 4-regular line graph. We con-struct a graph H from G as follows. V ( H ) contains a clique { e , . . . , e | E ( G ) |− } where each e i , 0 ≤ i ≤ | E ( G ) | −
1, is associated with a distinct edge of G . V ( H ) contains an stable set { e (cid:48) , . . . , e (cid:48)| E ( G ) |− } such that each e (cid:48) i is adjacentto all e j with j (cid:54) = i and j (cid:54) = i + 1 mod | E ( G ) | . V ( H ) contains an stableset { v , . . . , v | V ( G ) |− } , where each v i , 0 ≤ i ≤ | V ( G ) | −
1, is associated witha distinct vertex of G . Each e j ∈ { e , . . . , e | E ( G ) |− } is adjacent to vertices v r , v s ∈ { v , . . . , v | V ( G ) |− } such that e j = v r v s . V ( H ) contains an stable set P comprising ( | V ( G ) | + | E ( G ) | )( | E ( G ) | + 1) pendant vertices such that eachvertex of { v , . . . , v | V ( G ) |− } ∪ { e (cid:48) , . . . , e (cid:48)| E ( G ) |− } is adjacent to | E ( G ) | + 1 pen-dant vertices. By construction, H is (2 ,
1) and chordal.We claim that T t ( H ) = | E ( G ) | + 3 if and only if χ ( G ) = 3. Consider a 3-coloring c of G . Obtain a k -total tessellation cover of H with k = | E ( G ) | +3 as fol-lows. Assign colors in { ,. . . , | E ( G ) |} to the vertices of the clique { e ,. . . ,e | E ( G ) |− } .Assign to vertex e (cid:48) i , for 1 ≤ i ≤ | E ( G ) | , the same color of the vertex e i .For 0 ≤ i ≤ | E ( G ) | −
1, the tile with vertices { e (cid:48) i } ∪ { e j | j (cid:54) = i and j (cid:54) = i + 1 mod | E ( G ) |} is in the tessellation with label i + 2 mod | E ( G ) | . Notethat if two vertices v i and v k of G are not adjacent, then the cliques { v i } ∪{ e j | v i is endpoint of e j in G } and { v k } ∪ { e j | v k is endpoint of e j in G } aredisjoint. Thus, the tile with vertices { v i } ∪ { e j | v i is endpoint of e j in G } isin the tessellation with label c ( v i ) + | E ( G ) | . Finally, greedily assign colors andlabels to the remaining vertices and edges of H . Consider a total tessellation A. Abreu et al. cover of H with k = | E ( G ) | + 3 labels. Note that we require | E ( G ) | tessellationsto cover the edges between the vertices { e , . . . , e | E ( G ) |− } ∪ { e (cid:48) , . . . , e (cid:48)| E ( G ) |− } in any total tessellation cover of H . Moreover, a tile in each of those | E ( G ) | tessellations contains | E ( G ) | − { e , . . . , e | E ( G ) |− } . Sinceeach tile { v i } ∪ { e j | v i is endpoint of e j in G } , for 0 ≤ i ≤ | V ( G ) | −
1, con-tains four vertices of the clique { e , . . . , e | E ( G ) |− } , there are only three tessella-tion labels used by the tiles { v i } ∪ { e j | v i is endpoint of e j in G } , for 0 ≤ i ≤| V ( G ) | −
1. Moreover, if two vertices v i and v k are adjacent in G , then the tiles { v i } ∪ { e j | v i is endpoint of e j in G } and { v k } ∪ { e j | v k is endpoint of e j in G } share a vertex e j = v i v k in H and they are tiles belonging to different tessella-tions. Hence, we obtain a 3-coloring c of G as follows. Assign the label of the tile { v i } ∪ { e j | v i is endpoint of e j in G } to the color of v i in c .Therefore, G has a 3-coloring if and only if H has a total tessellation coverwith | E ( G ) | + 3 labels. Note that k = is ( H ) + 1 = ω ( H ) + 3 = | E ( G ) | + 3. Theorem 4. k -total tessellability is N P -complete for chordal graphs.
We now show how to simulate a total staggered quantum walk on a graph G witha staggered quantum walk on its total graph Tot( G ). The total graph Tot( G )of G has V (Tot( G )) = V ( G ) ∪ E ( G ) and E (Tot( G )) = E ( G ) ∪ { u uw | u ∈ V ( G ), uw ∈ E ( G ) } ∪ { uv vw | uv ∈ E ( G ) and vw ∈ E ( G ) } . Let A = Tot( G ), A [ E ( G )] = Y and A [ V ( G )] = X . Subgraph Y is isomorphic to the line graph L ( G ) of G , and X is isomorphic to the original G . We define the clique K v = { v } ∪ { vw | vw ∈ E ( G ) } of A .Consider a total tessellation cover of a graph G . Define an associated tessel-lation cover of A as follows. Assign the labels of the edges of G to the respectiveedges of X and assign the color of each vertex v of G to the edges of A [ K v ]. Wesimulate the total staggered quantum walk on G with the staggered quantumwalk on A by considering the vertices of G as the corresponding vertices of X in A , and the edges of G as the corresponding vertices of Y in A . Fig. 2 depictsa total tessellation cover of a graph G and the associated tessellation cover of A = Tot( G ).Consider the walker located on a vertex a of G . If we apply the operator H j associated with the color of a , the walker hops to the edges incident to a (the edges ab and ac ). If we apply an operator associated with the label of anedge incident to a , the walker hops to the vertices in the tile of the tessellationof the same label that contains a (the vertices b and c ). The same happens byconsidering the walker located on a vertex a in X . If we apply the operator H j associated with the labels of the edges of A [ K a ], the walker hops to the vertices ab and ac of Y , and if we apply the operator associated with the label of anedge of X incident to a , the walker hops to the vertices b and c of X . Considerthe walker located on an edge ab of G . If we apply the operator associated withthe color of a (or b ), the walker hops to a (or b ) and to the edges incident to it. otal tessellation cover and quantum walk The same happens by considering the walker located on a vertex ab in Y . If weapply the operator associated with the labels of the edges of A [ K a ] (or A [ K b ]),the walker hops to vertices of K a (or K b ). Otherwise, the walker stays put inboth G and A . Gabd e fc Ha b c d e fab ac bc bd be ce cf de efA
Fig. 2.
Total tessellation cover of a graph G and the associated tessellation cover of A . We have defined the total tessellation cover on a graph G and have used thisconcept to define the total staggered quantum walk model. This work strengthensthe connection between quantum walk and graph coloring.We have established examples of graphs for which T t ( G ) reaches the boundsof Section 2. We leave as an open problem to search for graphs with at least3 vertices satisfying T t ( G ) = 3 T ( G ) and T t ( G ) > χ ( G ). Moreover, it would beinteresting to define graph classes with T t ( G ) = T ( G ) = k for k ≥
4, since for k = 3 the only such graphs are the odd cycles C n with n ≡ -total tessellability is N P -complete for planargraphs satisfying is ( G ) + 1 = ω ( G ) = 4. This is important since the hardnessof k -edge colorability and k -total colorability for planar graphs arestill open. On the other hand, we know that planar graphs with large maximumdegree have edge and total colorings as small as possible [19, 20]. We leave asan open problem to find a threshold for T t ( G ) for which all planar graphs areType II.Table 1 summarizes the computational complexities of edge-colorability cf. [17], total colorability cf. [21], tessellability cf. [8], and total tes-sellability . These four problems are in P when restricted to complete graphs,star graphs and trees, whereas for triangle-free graphs, the four problems are N P -complete. We leave as an open problem to find a graph class for which total colorability is N P -complete and total tessellability is in P .We have not identified this class because all known N P -completeness proofs of total colorability are restricted to graph classes with χ t ( G ) = T t ( G ). References
1. R. Portugal, R. A. M. Santos, T. D. Fernandes, D. N. Gon¸calves, The staggeredquantum walk model, Quantum Inf. Process. 15 (1) (2016) 85–101.0 A. Abreu et al. χ (cid:48) ( G ) T ( G ) χ (cid:48) ( G ) χ t ( G ) χ (cid:48) ( G ) T t ( G )[2 | V ( G ) | , G c ] P N P -c G ∪ K ∆ ( G )+1 , ∆ even P N P -c [2 | V ( G ) | , G c ] P N P -cLine of Bipar-tite
N P -c P G ∪ K ∆ ( G )+1 , ∆ odd N P -c P Line of Bipar-tite, ω ( G ) ≥ N P -c P T ( G ) X t ( G ) T ( G ) T t ( G ) χ t ( G ) T t ( G )Bipartite P N P -c Bipartite
P N P -c G ∪ K ∆ ( G )+1 , ∆ odd P N P -c[2 | V ( G ) | , G c ] N P -c P G ∪ K ∆ ( G ) N P -c P Open
N P -c P Table 1.
Computational complexities of parameters χ (cid:48) ( G ) , χ t ( G ) , T ( G ), and T t ( G ).2. D. West, Introduction to Graph Theory, Pearson, 2000.3. R. Portugal, Quantum Walks and Search Algorithms, Springer, 2013.4. S. Venegas-Andraca, Quantum walks: a comprehensive review, Quantum Inf. Pro-cess. 11 (5) (2012) 1015–1106.5. A. Ambainis, Quantum search algorithms, ACM SIGACT News 35 (2004) 22–35.6. R. Portugal, Establishing the equivalence between Szegedy’s and coined quantumwalks using the staggered model, Quantum Inf. Process. 15 (4) (2016) 1387–1409.7. M. Szegedy, Quantum speed-up of Markov chain based algorithms, Proceedings ofthe 45th Symposium on Foundations of Computer Science (2004) 32–41.8. A. Abreu, L. Cunha, T. Fernandes, C. de Figueiredo, L. Kowada, F. Marquezino,D. Posner, R. Portugal, The graph tessellation cover number: extremal bounds,efficient algorithms and hardness, Proceedings of the 13th Latin American Theo-retical Informatics Symposium, LNCS 10807 (2018) 1–13.9. A. Abreu, L. Cunha, C. de Figueiredo, L. Kowada, F. Marquezino, D. Posner,R. Portugal, The graph tessellation cover number: chromatic bounds, efficient al-gorithms and hardness, Theor. Comput. Sci. 801 (2020) 175–191.10. D. P. Koreas, The NP-completeness of chromatic index in triangle free graphs withmaximum vertex of degree 3, Appl. Math. Comput. 83 (1) (1997) 13–17.11. D. Posner, C. Silva, R. Portugal, On the characterization of 3-tessellable graphs,Proceedings Series of the Brazilian Society of Computational and Applied Mathe-matics, CNMAC 6 (2) (2018) 1–7.12. A. Abreu, L. Cunha, C. de Figueiredo, L. Kowada, F. Marquezino, D. Posner,R. Portugal, The tessellation cover number of good tessellable graphs, Submittedto Theor. Comput. Sci. arXiv:1908.10844 (2019) 1–14.13. P. Duchet, Repr´esentations, noyaux en th´eorie des graphes et hypergraphes, Ph.D.thesis, Univ. Paris 6 (1979).14. C. J. McDiarmid, A. S´anchez-Arroyo, Total colouring regular bipartite graphs isNP-hard, Discrete Math. 124 (1994) 155–162.15. V. F. Santos, D. Sasaki, Total coloring of snarks is NP-complete, Matem´aticaContemporˆanea 44 (2015) 1–10.16. M. Gr¨otschel, L. Lov´asz, A. Schrijver, The ellipsoid method and its consequencesin combinatorial optimization, Combinatorica 1 (2) (1981) 169–197.17. R. C. Machado, C. M. de Figueiredo, K. Vuˇskovi´c, Chromatic index of graphs withno cycle with a unique chord, Theor. Comput. Sci. 411 (7–9) (2010) 1221–1234.18. M. R. Garey, D. S. Johnson, Computers and Intractability: A Guide to the Theoryof NP-Completeness, W. H. Freeman Co, 1979.19. R. Cole, (cid:32)L. Kowalik, New Linear-Time Algorithms for Edge-Coloring PlanarGraphs, Algorithmica 50 (3) (2008) 351–368. otal tessellation cover and quantum walk Appendix
Planar Graphs - Detailed proof of Theorem 3
The computational complexity of total coloring for planar graphs is an openproblem. On the other hand, we show in this section that 4 -total tessella-bility for planar graphs is
N P -complete. Since in this proof we use a genericgraph G such that is ( G ) + 1 = ω ( G ) = 4, we also prove that deciding whethera graph has both T ( G ) = is ( G ) + 1 and T ( G ) = ω ( G ) is N P -complete even ifrestricted to planar graphs.
Lemma 3.
Let G be the graph with V ( G ) = { a , b , c , d } ∪ { a , b , c , d } ∪{ a , b , c , d } , where { a , b , c , d } is a maximal clique and { a , a , a } , { b , b , b } , { c , c , c } , and { d , d , d } are triangles. Any total tessellation cover of G withfour labels has the following property: The edges of three triangles are tiles on asame tessellation and the edges of the remaining triangle are a tile on a differenttessellation.Proof. The proof follows after analyzing all possibilities of total tessellation cov-ers with four labels. (cid:117)(cid:116) a b c d a a b b c c d d Fig. 3.
A 4-total tessellation cover of the graph of Lemma 3.
Fig. 3 shows an example of a total tessellation cover of graph G describedin Lemma 3. The edges of three triangles must have the same color (red) andthe fourth triangle must have a different color (blue). In the first gadget of theFig. 4, the two external triangles’ edges must receive the same label, since thetwo internal triangles shares a vertex and they must receive different labels. Inthe second gadget, since the internal triangles’ edges must receive the same labelthe external triangle’ edges must receive different labels. Lemma 4.
Any total tessellation cover of the graph G of Fig. 6 with four labelshas the following property: The triangles T and T are tiles in a same tessel-lation, and the triangles T and T are tiles in a same tessellation, which isdifferent from the tessellation that contains T and T .Proof. Since there are three maximal cliques incident to the vertices b (resp. c and e ), they are three tiles on different tessellations. Therefore, the three trianglesof the Hajos subgraph of G are tiles on different tessellations. The Equal Gadgethas its tiles incident to the vertices a and d on a same tessellation. otal tessellation cover and quantum walk Fig. 4.
Equal Gadget: edges of its two external triangles are covered by 3-tiles of a sametessellation in a 4-total tessellation cover. NotEqual Gadget: edges of its two externaltriangle are covered by 3-tiles of different tessellations in a 4-total tessellation cover.
Fig. 5.
Duplicator Gadget: it forces the five external triangles’ edges to have the samelabel. Moreover, if the label of the triangle’ edges is a , then its vertices have the nexttwo consecutive labels a + 1 and a + 2 modulo 4 available to the vertices of four ofthese five triangles in a 4-total tessellation cover.4 A. Abreu et al. abd e cT T T T fb' c'e'abd e cT T T T fb' c'e' (a)(b) Fig. 6.
Shifter Gadget: it shifts two crossing tiles in different tessellations such thatthe tiles get from one side to the other side without crossing edges and maintainingtheir 3-tiles tessellations in a 4-total tessellation cover. otal tessellation cover and quantum walk Assume that the color of the vertex a is equal to the color of the vertex d .Since the color of vertex a is the same of the color of vertex d and the tiles ofthe Equal Gadget have a same tessellation different from the label of the colorof a and d , it implies that T is a tile on the same tessellation of the trianglewith vertices { b, d, e } and that T is a tile on the same tessellation of the trianglewith vertices { a, b, c } . Note that the colors of the vertices a (resp. d ) and b aredifferent and that they are also different from the labels of T and T . Thereforethe color of the vertices c and e must be the same labels of the tessellations of T and T . Now, the triangle { c, e, f } and the vertex f must receive two labelsdifferent from the labels used by the triangles { a, b, c } and { b, d, e } . This impliesthat the triangles T and T are tiles with the same labels of the triangles T and T . Since T share a vertex with T and T share a vertex with T , thetriangles T and T are tiles on a same tessellation and the triangles T and T are also tiles on a same tessellation different from the tessellation of T and T .Therefore, if there is a total tessellation cover of G with 4 labels, the proof ofthe theorem holds. A total tessellation cover of G with 4 labels is depicted inFig. 6 (a). Note that we obtain total tessellation covers of G with all possiblecombinations of two distinct labels of the four labels for T and T by replacingthe color classes of G by the desired labels.Assume that the color of the vertex a is different from the color of the vertex d . For the sake of contradiction assume we use only two labels to the colors of a , d and the tiles of the triangles { a, b, c } and { b, d, e } . This implies that b receives athird label different from these two, and that there is only one available label tothe color of the vertices c and e , a contradiction. We also cannot use four differentlabels to the vertices a , d and the triangles { a, b, c } and { b, d, e } or there wouldbe no available label to the triangles of the Equal Gadget. Therefore, we havethree different labels used in the colors of the vertices a , d and the labels ofthe tiles of the triangles { a, b, c } and { b, d, e } . This implies that the color of thevertex b and the tiles of the Equal Gadget receive the same label. The color ofthe vertices a , b , and the label of the tile of the triangle { a, b, c } are differentfrom the labels of T . This implies that the color of c is equal to the label of thetile of T . The same holds for the vertex e and the label of the tile of T . Nowthe color of the vertex f and the label of the tile of the triangle { c, e, f } mustbe different from the colors of c and e (i.e., the label of the tiles of T and T ).This implies that the label of the tiles T and T are the same labels of the tiles T and T . Since T share a vertex with T and T share a vertex with T , thetriangles T and T are tiles on a same tessellation and the triangles T and T are also tiles on a same tessellation different from the tessellation of T and T .Therefore, if there is a total tessellation cover of G with 4 labels, the proof ofthe theorem holds. A total tessellation cover of G with 4 labels is depicted inFig. 6 (b). Note that we obtain total tessellation covers of G with all possiblecombinations of two distinct labels of the four labels for T and T by replacingthe color classes of G by the desired labels. (cid:117)(cid:116) Theorem 3 -total tessellability is N P -complete for planar graphs.
Proof.
Let G be an instance of 3 -colorability of planar graphs with degreeat most four [18]. Add a universal vertex u to G so that G ∨ { u } has a 4-coloringif and only if G has a 3-coloring. We create a planar graph H from G ∨ { u } as follows. We replace each vertex of G ∨ { u } by a Duplicator Gadget with thedegree of the vertex duplication in H . We replace each edge of G ∨ { u } by aNotEqual Gadget connecting the related triangles of the Duplicator Gadgetsof the endpoints of the edge in H . The only crossing edges in H are from thetriangles of the universal vertex and the triangles of the other Duplicator Gadgetthat have labels different from the one of the universal Gadget. We replace thesecrossing tiles with the Shifter Gadget.We claim that the resulting planar graph H has a 4-total tessellation coverif and only if the graph G has a 3-coloring.Consider a 4-total tessellation cover of H . If two vertices are adjacent in G ∨ { u } , then the NotEqual Gadget forces the tiles of the external trianglesof the respective Duplicator Gadgets of these two adjacent vertices to be ondifferent tessellations. Therefore, we obtain a 4-coloring of G ∨ { u } by assigningthe color of a vertex as the label of the tile of the external triangles of theDuplicator Gadget related to that vertex.Consider a 4-coloring f of G ∨{ u } . We obtain a 4-total tessellation cover of H as follows. Assign each tile of the external triangles of the Duplicator Gadget tothe tessellation related to the color the vertex received in f . Label the remainingvertices and edges as described in Figure 5 by rotating the color classes labelsto obtain the desired label.Since we obtain the total tessellation cover of the Duplicator Gadget byrotating the color classes, we have that the label of a external triangle and avertex of the degree two is related to consecutive colors. Therefore, if the labelof the tile of the external triangle is 1 (resp. 2, 3, and 4), then there are twovertex of degree two in this external triangle with colors 2 and 3 (resp. 3 and 4, 4and 1, 1 and 2). Now, for any two different tessellations of the tiles of the externaltriangles, we select one vertex of degree two of each so that we do not use allfour labels in these two vertex and in the two tiles of the external triangles. ByLemma 3, there is a total tessellation cover with four labels of the NotEqualGadget if we do not use all four labels on the two tiles of its external trianglesand the two vertices of the K of that external triangles.We obtain a total tessellation cover with 4 labels of the Shifter Gadgets asdescribed in Lemma 4. Note that, as depicted in Fig. 6, the two consecutive EqualGadgets connected to the external triangles T (resp. T , T , and T ) allow usto assign colors to the vertices of the Shifter Gadgets so that the vertices of thelast of their external triangles have the same colors of the vertices of the externaltriangles of the Duplicator Gadgets that they are related.) allow usto assign colors to the vertices of the Shifter Gadgets so that the vertices of thelast of their external triangles have the same colors of the vertices of the externaltriangles of the Duplicator Gadgets that they are related.