A Note on Colourings of Connected Oriented Cubic Graphs
aa r X i v : . [ c s . D M ] M a y COLOURINGS OF ORIENTED CONNECTED CUBIC GRAPHS
Christopher Duffy Department of Mathematics and Statistics, University of Saskatchewan, CANADA
Abstract.
In this note we show every orientation of a connected cubic graph admits anoriented 8-colouring. This lowers the best-known upper bound for the chromatic num-ber of the family of orientations of connected cubic graphs. We further show that everysuch oriented graph admits a 2-dipath 7-colouring. These results imply that either theoriented chromatic number for the family of orientations of connected cubic graphs equalsthe 2-dipath chromatic number or the long-standing conjecture of Sopena [
Journal of GraphTheory 25:191-205 1997 ] regarding the chromatic number of orientations of connected cubicgraphs is false. Introduction and Preliminary Notions An oriented graph is a simple graph equipped with an orientation of its edges as arcs.Equivalently, an oriented graph is an antisymmetric loopless digraph. We call an orientedgraph G properly subcubic when its underlying simple graph, denoted U ( G ), has maximumdegree three and a vertex of degree at most two. The degree of a vertex in G is its degree in U ( G ). We call an oriented graph G connected when U ( G ) is connected. For uv, vw ∈ A ( G ),we say uvw is a 2 -dipath ; v is between u and w ; and v is the centre of the 2-dipath uvw . Forgraph theoretic notation and terminology not defined herein, we refer the reader to [1].Let G and H be oriented graphs. There is a homomorphism of G to H when there exists φ : V ( G ) → V ( H ) so that uv ∈ A ( G ) implies φ ( u ) φ ( v ) ∈ A ( H ). We call φ a homomorphism or an oriented | V ( H ) | -colouring of G . The oriented chromatic number of an oriented graph G , denoted χ o ( G ), the least integer k such that there is a homomorphism of G to an orientedgraph with k vertices. That is, χ o ( G ) is the least integer k so that there exists an oriented k -colouring of G . For F , a family of oriented graphs, we define χ o ( F ) to be the least integer k such that χ o ( F ) ≤ k for all F ∈ F .Equivalently one may define an oriented k -colouring as a labelling c : V ( G ) → { , , . . . , k − } such that:(1) c ( u ) = c ( v ) for all uv ∈ A ( G ); and(2) if uv, xy ∈ A ( G ) and c ( u ) = c ( y ), then c ( v ) = c ( x ).Such a labeling implicitly defines a homomorphism to an oriented graph H with vertex set { , , . . . , k − } where ij ∈ A ( H ) when there is an arc uv ∈ A ( G ) such that φ ( u ) = i and φ ( v ) = j . Condition (1) ensures H is loopless and condition (2) ensures H is antisymmetric.In the case v = x , condition (2) implies that in an oriented colouring any three verticesforming a directed path (i.e., a 2-dipath) receive distinct colours. As such one may bound Research supported by the Natural Science and Research Council of Canada he oriented chromatic number of an oriented graph by considering colourings that assigndistinct colours to vertices at directed distance at most two.Let G be an oriented graph. A t -colouring of G is a labelling c : V ( G ) →{ , , . . . , t − } such that:(1) c ( u ) = c ( v ) for all uv ∈ A ( G ); and(2) if uv, vw ∈ A ( G ), then c ( u ) = c ( w ).The 2 -dipath chromatic number of an oriented graph G , denoted χ d ( G ), is the least integer t such that G admits a 2-dipath t -colouring. For F , a family of oriented graphs, χ d ( F ) isdefined to be the least integer t such that χ d ( F ) ≤ t for all F ∈ F . Since every orientedcolouring is 2-dipath colouring it follows that for any oriented graph H we have χ o ( H ) ≥ χ d ( H ).Consider an oriented graph G for which every pair of vertices is either adjacent or the endsof a 2-dipath. The definitions of oriented colouring and 2-dipath coloring imply χ o ( G ) = χ d ( G ) = | V ( G ) | . We call such oriented graphs oriented cliques .The notion of 2-dipath colourings was introduced in [5, 10]. We refer the reader to [9] for acomprehensive survey on homomorphism and colourings of oriented graphs.Let F be the set of orientations of cubic graphs. Let F C be the set of orientations ofconnected cubic graphs. In [3] the authors show χ o ( F c ) ≤
9. They further show that thisbound may be improved to 8 when restricted to those oriented connected cubic graphs thathave a source or a sink. Central to these results are homomorphisms to a particular class ofCayley digraphs, namely Paley tournaments. Let q be a prime power congruent to 3 modulo4. The Paley tournament on q vertices , denoted QR q , is the tournament with vertex set { , , , . . . , q − } , where uv ∈ A ( G ) when v − u is a non-zero quadratic residue modulo q . Theorem 1. [3] If G is a connected properly subcubic oriented graph with no degree sourcevertex adjacent to a degree sink vertex, then G → QR . Our work proceeds as follows. In Section 2 we show that if G is an orientation of a connectedcubic graph, then χ o ( G ) ≤
8, regardless of the presence of sources and sinks. This improvesthe best known upper bound for the oriented chromatic number of orientations of connectedcubic graphs. In Section 3 we show that every oriented cubic graph admits a 2-dipath 7-colouring. Together these results point to a new line of attack on the long-standing conjectureof Sopena [8] regarding the oriented chromatic number of connected cubic graphs. This isdiscussed further in Section 4.2.
Oriented Colourings of Orientations of Connected Cubic Graphs
For convenience we provide the following result regarding QR . Lemma 2. [6](1)
The tournament QR is vertex transitive and arc transitive. For yz ∈ A ( QR ) , there exists a pair of distinct vertices x, x ′ ∈ V ( QR ) so that xyz and x ′ yz are directed cycles in QR . We begin with technical lemma.
Lemma 3. If G is an oriented cubic graph with no source and no sink, then • G contains a vertex with out-degree whose out-neighbours both have in-degree ; or • G contains a vertex of out-degree that has an out-neighbour of in-degree and anin-neighbour of out-degree .Proof. Without loss of generality assume G is connected. We partition the vertices of G based on their out-degree. Let V + be the set of vertices of G with out-degree 2 and V − be the set of vertices in G with in-degree 2. If G [ V + ] has a vertex with out-degree 0, thensuch a vertex has two out-neighbours in V − and so is a vertex with out-degree 2 whoseout-neighbours both have in-degree 2. Otherwise, assume every vertex in V + has out-degreeat least 1 in G [ V + ].As every vertex in G [ V + ] has out-degree at least 1 in G [ V + ], the oriented graph G [ V + ]contains at least one directed cycle, C . Note that such a cycle is necessarily induced; thehead of a chord in such a cycle would have in-degree 2. Let x be a vertex of this cycle, andconsider xx ′ ∈ A ( G ) such that x ′ is not contained in C . If x ′ ∈ V − , then x is a vertex ofout-degree 2 that has an out-neighbour of in-degree 2 and an in-neighbour of out-degree 2.And so assume x ′ ∈ V + .Consider a maximal directed walk W in G beginning with xx ′ so that at most a single vertexof G appears twice in W . If W contains a vertex of in-degree 2, then the first arc of W that has its tail in V + and its head in V − contains a vertex of out-degree 2 that has anout-neighbour of in-degree 2 and an in-neighbour of out-degree 2. So assume such a walkcontains only vertices with out-degree 2. That is, W is contained wholly in G [ V + ].As each vertex in G [ V + ] has positive out-degree and W is maximal, W contains a directedcycle C ′ . We claim C and C ′ contain no common vertices. If y ∈ C and y ∈ C ′ then byconstruction there is a directed path Q from x to y in G [ V + ] that begins with the arc xx ′ .Since x ′ / ∈ C and y ∈ C there is a first vertex y ′ ∈ Q such that the in-neighbour of y ′ is in Q but not in C . However y ′ also has an in-neighbour in C . This contradicts that y ′ ∈ V + Therefore C and C ′ contain no common vertices.As W begins with x and contains a vertex from C ′ , there is a directed path P (which isa subwalk of W ) from x to a vertex in C ′ . The last vertex on this path, i.e., the first onecontained in C ′ , has both an in-neighbour in P and an in-neighbour in C ′ . Such a vertexis contained in V − . This contradicts that W is wholly contained within G [ V + ]. Thus W contains a vertex of in-degree 2 and the proof is complete. (cid:3) Lemma 4. If G is an oriented connected cubic graph with no source and no sink and U ( G ) contains a triangle, then χ o ( G ) ≤ . roof. Let G be an oriented connected cubic graph with no source and no sink so that thevertices u, v, w induce a triangle in U ( G ) and uv ∈ A ( G ). Without loss of generality, thereare two possible orientations: uvw is a directed cycle or uw, vw ∈ A ( G ).By Theorem 1, there is a homomorphism φ : G − uv → QR . If φ ( u ) = φ ( v ), then modifying φ to let φ ( u ) = 7 yields an oriented 8-colouring of G . And so we may assume φ ( u ) = φ ( v ).As QR is arc transitive, we may assume φ ( v ) = 0 and φ ( u ) = 1. (Note that if φ ( v ) = 1and φ ( u ) = 0, then φ : G → QR ). Let u ′ = w and v ′ = w so that u ′ and v ′ are respectivelyneighbours of u and v . If uu ′ ∈ A ( G ) or φ ( u ′ ) = 0, then modifying φ such that φ ( u ) = 7yields an oriented 8-colouring of G . And so we may assume u ′ u ∈ A ( G ) and φ ( u ′ ) = 0.Similarly, we may assume vv ′ ∈ A ( G ) and φ ( v ′ ) = 1. Let w ′ / ∈ { u, v } be a neighbour of w . Case I: uvw is a directed cycle in G . Consider modifying φ in one of three ways:(1) φ ( u ) = 2 , φ ( v ) = 4;(2) φ ( u ) = 2 , φ ( v ) = 6; or(3) φ ( u ) = 4 , φ ( v ) = 6.Note that in each case φ ( u ) φ ( u ′ ) , φ ( v ′ ) φ ( v ) , φ ( u ) φ ( v ) ∈ A ( QR ). By part (2) of Lemma2, each of these three possibilities allow us to modify φ ( w ) in two possible ways so that φ ( w ) φ ( u ) , φ ( v ) φ ( w ) ∈ A ( QR ):(1) φ ( w ) = 1 , φ ( w ) = 0 ,
1; or(3) φ ( w ) = 0 , QR has both an in-neighbour and an out-neighbour in the set { , , , } . As such, regardless of the orientation of the edge ww ′ and the value of φ ( w ′ ), wecan choose φ ( u ) , φ ( v ) and φ ( w ) so that φ : G → QR is a homomorphism. This completesCase I. Case II: uw, vw ∈ A ( G ) . Since G has no source or sink vertex, ww ′ ∈ V ( G ). Proceedingas in Case I, we note that φ ( u ) and φ ( v ) can be modified so that φ ( w ′ ) / ∈ { φ ( u ) , φ ( v ) } and φ ( u ) φ ( v ) ∈ A ( QR ). Modify φ so that φ ( u ) = φ ( w ′ ) , φ ( v ) = φ ( w ′ ) and φ ( w ) = 7. This yieldsan oriented 8-colouring of G . (cid:3) Lemma 5. If G is an oriented connected cubic graph with no source and no sink and U ( G ) is triangle free, then χ o ( G ) ≤ .Proof. Let G be an oriented connected cubic graph with no source and no sink so that U ( G )is triangle free. Consider first the case that G has a cut arc, say uv . Let G u be the componentof G − uv that contains u . Similarly, let G v be the component of G − uv that contains v . ByTheorem 1, each of G u and G v admit a homomorphism to QR . Further, by Lemma 2 thereexist homomorphisms φ u : G u → QR and φ v : G v → QR so that φ u ( u ) = 0 and φ v ( v ) = 1.Combining φ u and φ v yields a homomorphism of G to QR . Thus χ o ( G ) ≤ G has no cut arc. By Lemma 3, in G there is an arc from a vertex ofout-degree 2 to a vertex of in-degree 2. Let x and u be such vertices so that xu ∈ A ( G ). Let be the out-neighbour of u . Let w = x be an in-neighbour of u . Let z be the in-neighbourof x . Let y = u be an out-neighbour of x . By Lemma 3 we may choose x and u so that z has out-degree 2 or y has in-degree 2. Note that as U ( G ) is triangle free, z and y are notadjacent. Construct G ′ from G by removing x and adding the arc yz . The oriented graph G ′ is properly subcubic. We further note that as xu is not a cut arc, the oriented graph G ′ is connected.Recall that in G vertex z has out-degree 2 or y has in-degree 2. If y is a source vertex in G ′ ,then y has out-degree 2 in G . Therefore z has out-degree 2 in G and so is not a sink vertexin G ′ . Similarly, if z is a sink vertex in G ′ , then y is not a source vertex in G ′ . As G has nosource vertex and no sink vertex, exactly one of the following is true: (1) G ′ has no sourceor sink vertex of degree 3; (2) G ′ has a sink vertex of degree 3, namely z , and no sourcevertex of degree 3, or (3) G ′ has a source vertex of degree 3, namely y , and no sink vertexof degree 3. Thus G ′ , has no degree 3 source vertex adjacent to a degree 3 sink vertex. ByTheorem 1, there exists a homomorphism φ : G ′ → QR . As QR is vertex transitive, wemay assume φ ( v ) = 0. By part (2) of Lemma 2, φ can be extended to include x so that φ ( x ) = 0. Note that φ ( w ) = 0 as there is a 2-dipath from w to v and φ ( v ) = 0. Recoloring u so that φ ( u ) = 7 gives an oriented 8-colouring of G . (cid:3) Theorem 6. If G is an oriented connected cubic graph, then χ o ( G ) ≤ .Proof. If G has a source or a sink vertex, then the result follows from Corollary 4.9 in [3]. If G has no source and no sink and U ( G ) contains a triangle, then the result follows by Lemma4. Otherwise, G has no source and no sink and U ( G ) contains no triangle. The result thenfollows by Lemma 5. (cid:3) Figure 1 in [3] gives an oriented clique on 7 vertices whose underlying graph has maximumdegree 3. Thus χ o ( F C ) ≥
7. Combining this with the statement of Theorem 6 yields thefollowing.
Theorem 7.
For F C , the family of orientations of connected cubic graphs, we have ≤ χ o ( F C ) ≤ .
3. 2 -dipath Colourings of Orientations of Cubic Graphs
For an oriented graph G , let G be the simple undirected graph formed from G by firstadding an edge between any pair of vertices at directed distance exactly 2 in G (i.e., verticesat the end of an induced 2-dipath) and then changing all arcs to edges. One easily observes χ ( G ) = χ d ( G ). Thus we approach our study of χ d ( F ) by examining the chromaticnumber of graphs of the form G for G ∈ F . Let F = { G | G ∈ F } . In [2] the authorsestablish ω ( F ) = 7. Thus χ d ( F ) ≥
7. Here we show χ d ( F ) = 7. Lemma 8. If G is an orientation of a cubic graph, then G is -regular or G has a vertexof degree at most . roof. Let G be an orientation of a cubic graph with n vertices. We show G has averagedegree 7. Every edge in G corresponds to an arc in G or to an induced 2-dipath in G . Everyvertex in G is the centre vertex of at most two induced 2-dipaths. Therefore there are atmost 2 n induced 2-dipaths in G . And so | E ( G ) | ≤ n + 2 n = n . Thus G has averagedegree 7. (cid:3) Lemma 9. If G is an orientation of a cubic graph and G is -regular, then χ ( G ) ≤ .Proof. By Brooks’ Theorem, it suffices to show G is not a complete graph. If G is acomplete graph, then it an oriented clique with at least 8 vertices. However this contradictsthe statement of Proposition 3.3 in [3]. (cid:3) Lemma 10. If G is an orientation of a connected cubic graph and G has a source vertexadjacent to sink vertex, then χ d ( G ) ≤ .Proof. Let G be an orientation of a connected cubic graph. Let s , s , . . . s ℓ and t , t , . . . , t ℓ respectively be source and sink vertices so that s i t i ∈ A ( G ) for all 1 ≤ i ≤ ℓ . (Note that s , s , . . . , s ℓ may not all be distinct vertices. Similarly t , t , . . . , t ℓ may not all be distinctvertices.) Form G ′ from G by first deleting each arc s i t i and then adding vertex x i sothat s i x i , x i t i ∈ A ( G ′ ). Notice that for any pair u, v ∈ V ( G ), if u and v are at directeddistance at most 2 in G , then u and v are at directed distance at most 2 in G ′ . Therefore χ d ( G ) ≤ χ d ( G ′ ). The oriented graph G ′ is a connected properly subcubic oriented graphwith no degree 3 source adjacent to a degree 3 sink. And so by Theorem 1, it follows G ′ → QR . Thus χ o ( G ′ ) ≤ χ d ( G ) ≤ (cid:3) Lemma 11. If G is an orientation of a connected cubic graph and G with no source vertexadjacent to sink vertex, then χ d ( G ) ≤ .Proof. We proceed by contradiction. Consider, G , an orientation of a connected cubic graphwith no source vertex adjacent to sink vertex so that χ d ( G ) >
7. Clearly G QR , asotherwise χ o ( G ) ≤
7. However we note that by Theorem 1, every proper subgraph of G admits a homomorphism to QR .If G is 7-regular, then the result follows by Lemma 9. If G is 6-degenerate, then χ ( G ) ≤ G is not 6-degenerate nor 7-regular. Therefore G containsan induced subgraph with minimum degree 7. Let C be the vertices of a maximum suchinduced subgraph of G . Let H = V ( G ) \ C . By Lemma 8, G has a vertex of degree atmost 6 and so C = V ( G ) and H = ∅ .Since C is maximum, there is an ordering of the elements of H : x , x , x , . . . , x ℓ so that forall 1 ≤ i ≤ ℓ vertex x i has degree at most 6 in the subgraph of G induced by the verticesof C together with x , x , . . . , x i − . We see then that if G [ C ] admits a 7-colouring, such acolouring can be extended to a 7-colouring of G . As this would be a contradiction, we mayassume χ ( G [ C ]) > G [ C ] arise from arcs in G [ C ] and from 2-dipaths in G whose ends are in C . Forthese latter edges, it is possible the the centre vertex of such a 2-dipath is not containedin C . Let B C = { w ∈ H | w is between a pair of vertices in C } . That is, B C is the set ofvertices of G that are not in C and are the centre vertex of a 2-dipath in G whose ends are n C . Consider the subgraph of G , G C , formed from G [ C ] by adding the vertices of B C and all arcs with exactly one endpoint in C and one end point in B C . Since U ( G ) is cubic,every proper subgraph U ( G ) has components that are properly subcubic. Thus, if G C is aproper subgraph of G (i.e, if G C = G ), then every component of G C is a connected properlysubcubic oriented graph with no degree 3 source vertex adjacent to a degree 3 sink vertex.By Theorem 1 there exists a homomorphism φ : G C → QR . Restricting φ to the verticesof C yields an 7-colouring of G [ C ]. This is a contradiction as χ ( G [ C ]) >
7. Therefore G C = G .Since G C = G it follows B C = H and G [ H ] is an independent set. We proceed to bound e C ,the number of edges in G [ C ]. Recall that edges in G [ C ] arise from arcs in G [ C ] and frominduced 2-dipaths in G , xyz , so that x, z ∈ C .The oriented graph G is cubic and so has | C | + | H | )2 arcs. Of these, exactly 3 | H | arcs haveexactly one end in H . Therefore G [ C ] has at most | C | + | H | )2 - 3 | H | edges that arise fromarcs in G [ C ].Since V ( G ) = C ∪ H , the set of induced 2-dipaths xyz ∈ G with x, z ∈ C can be partitionedinto those for which y ∈ H and those for which y ∈ C .Each vertex of H is the centre vertex of at most two induced 2-dipaths in G . Therefore thereare at most 2 | H | induced 2-dipaths xyz in G with x, z ∈ C so that y ∈ H .Since each vertex of C has degree at least 7 in G [ C ] and each vertex of G has at most sixvertices at distance exactly 2 in U ( G ), we observe G [ C ] has no isolated vertices. As eachvertex of C has degree at most 3 and at least 1 in G [ C ], each vertex y ∈ C is the centrevertex of at most deg G [ C ] − ≥ G [ C ]. Therefore there are at most X y ∈ C deg G [ C ] ( y ) − X y ∈ C deg G [ C ] ( y ) ! − | C | induced 2-dipaths in G [ C ]. To find the sum of the degrees of the vertices in G [ C ], we recall G is cubic and so the sum of the degrees in G is 3( | C | + | H | ). Discarding a vertex H fromthis count decreases this sum by exactly 6; each vertex in H is incident with 3 arcs and eachof these arcs has an endpoint in C . Therefore P y ∈ C deg G [ C ] ( y ) = 3( | C | + | H | ) − | H | . And sothere are at most 3( | C | + | H | ) − | H | − | C | = 2 | C | − | H | induced 2-dipaths in G [ C ]. As such there are at most 2 | C | − | H | induced 2-dipaths xyz in G with x, z ∈ C so that y ∈ C .Therefore e C ≤ | C | + | H | )2 − | H | + 2 | H | + 2 | C | − | H | = 7 | C | − | H | . Since H is non-empty, we conclude e C < | C | . On the other hand, we recall G [ C ] hasminimum degree 7. And so e C ≥ | C | , a contradiction. (cid:3) heorem 12. If G is an orientation of a cubic graph, then χ d ( G ) ≤ .Proof. It suffices to assume G is connected. The result follows directly from Lemmas 10 and11. (cid:3) Discussion
In an early investigation into the oriented chromatic number of orientations of connectedcubic graphs Sopena [8] conjectured χ o ( F C ) = 7. This conjecture has been verified for allorientations of connected cubic graphs with fewer than 20 vertices [7]. Here we have shown χ o ( F C ) ∈ { , } and χ d ( F ) = 7. Our results imply exactly one of the following must betrue:(1) χ o ( F C ) = 8 or(2) χ o ( F C ) = χ d ( F ) = 7.In other words, either Sopena’s conjecture is false or any orientation of a cubic graph with2-dipath chromatic number 7 also has oriented chromatic number 7. Notably, a statementanalogous to (2) is true for orientations of 2-regular graphs. To wit, χ o ( F C ) = χ d ( F ) = 5.(In fact χ o ( F ) = χ d ( F ) = 5 [9].) It remains to be seen if this is an artefact of the simplestructure of 2-regular graphs or due to some deeper connection between oriented colouringsand 2-dipath colourings of orientations of bounded degree graphs. In either case, the studyof oriented graphs that have oriented chromatic number equal to 2-dipath chromatic numberpresents a new line of attack for this long-standing open problem.In [5] the authors define, for each k >
1, an oriented graph H k with the property that G → H k if and only if χ d ( G ) ≤ k . An important next step in the study of homomorphismsof orientations of connected cubic graphs is the a study of the oriented chromatic number ofvarious subgraphs of H .Our results for the oriented chromatic number are limited to orientations of cubic graphsthat are connected. Recent work by Dybizba´nskia, Ochem, Pinlou and Szepietowskia [4]extend the work in [3] and show that all oriented cubic graphs admit an oriented 9-colouring.This result follows from construction of a 9-vertex universal target, based on QR , for thefamily of orientations of cubic graphs. Using similar methods, these authors provide newupper bounds for the oriented chromatic number of other families of orientations of boundeddegree graphs. It remains to be seen if Theorem 6 can be extended in a similar manner. References [1] JA Bondy and USR Murty. Graph theory, 2008.[2] Sandip Das, Swathy Prabhu, and Sagnik Sen. A study on oriented relative clique number.
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