On the existence and non-existence of improper homomorphisms of oriented and $2$-edge-coloured graphs to reflexive targets

We consider non-trivial homomorphisms to reflexive oriented graphs in which some pair of adjacent vertices have the same image. Using a notion of convexity for oriented graphs, we study those oriented graphs that do not admit such homomorphisms. We fully classify those oriented graphs with tree-width $2$ that do not admit such homomorphisms and show that it is NP-complete to decide if a graph admits an orientation that does not admit such homomorphisms. We prove analogous results for $2$-edge-coloured graphs. We apply our results on oriented graphs to provide a new tool in the study of chromatic number of orientations of planar graphs -- a long-standing open problem.


Introduction and Preliminaries
The main matter of this paper concerns homomorphisms of two objects that arise from graphs with no parallel edges -oriented graphs and 2-edge-coloured graphs. An oriented graph, − → G , arises from a graph G by assigning to each edge a direction to form an arc. Alternately an oriented graph is an anti-symmetric digraph. A 2-edge-coloured graph arises from a graph G by assigning a colour, red or blue, to each edge. Alternately, a 2-edge-coloured graph is a pair (G, c G ) where G = (V G , E G ) is a graph and c G : E G → {R, B}. In each case we refer to G as the underlying graph. As various portions of this work deals with reflexive and irreflexive graphs, oriented graphs and 2-edge-coloured graphs, we permit these objects to have loops unless otherwise specified. Reflexive 2-edge-coloured graphs are assumed to have a loop of each edge colour at every vertex. To simplify matters, we assume graphs herein are connected and have at least two vertices unless otherwise specified. For other notation and definitions not defined herein we refer to Murty and Bondy (2008).
For an oriented graph − → G with uv, vw ∈ A− → G with u = v and v = w we say that u, v, w is a 2-dipath and that v is the centre of the 2-dipath. For a 2-edge-coloured graph (G, c G ) with uv, vw ∈ E G with u = v and v = w we say that u, v, w is a 2-path, and that v is the centre of the 2-path. When c G (uv) = c G (vw) we say the 2-path is alternating. Otherwise we say the 2-path is monochromatic.
Let − → G and − → H be oriented graphs. We say there is a homomorphism of We say φ is a homomorphism and we write φ : − → G → − → H . Notice that φ induces a mapping from the arc set of − → G to the arc set of − → H . The study of oriented graph, and more generally, directed graph and graph homomorphisms has a rich history. Countless of pages of research articles, monographs and theses have been devoted to their study. Some of this focus comes by way of constraint satisfaction problems, which can be re-framed as questions of existence of certain directed graph homomorphisms. And so the computational complexity the − → H -colouring problem has been of particular interest. Bulatov (2017) and Zhuk (2017) independently verified the CSP Dichotomy Conjecture (see Feder and Vardi (1993)). This result implies the − → H -colouring problem can be classified as Polynomial or NP-complete based on the existence of a near-weak unanimity function for − → H . Another major research area resulting from oriented graph homomorphisms comes by way of the generalization of graph colouring to irreflexive oriented graphs. Courcelle (1994) first introduced these colourings as an example of a graph property expressible in the monadic second order logic of graphs. We return to this topic in Section 4 and so eschew definitions and further details here. We, however, point the eager reader to the survey by Sopena (2016).
For any oriented graphs H there exist non-trivial homomorphisms and even homomorphisms for which the induced mapping on the arc set uses no loop. For example, one can verify that there is a homomorphism of any orientation of a tree to the reflexive directed 3-cycle in which the induced mapping on the arc set uses no loop in the reflexive 3-cycle.
Let − → G be an oriented graph and let − → H be a reflexive oriented graph. Let φ : H be a homomorphism. The homomorphism φ can be classified as one of three types. We say φ is trivial when the induced mapping from the arc set of − → G to that of − → H maps every arc to the same loop. We say φ is improper when φ is not trivial and the induced mapping from the arc set of − → G to that of − → H maps at least one arc to a loop. Finally we say φ is proper when the induced mapping from the arc set of − → G to that of − → H maps no arc to a loop.
Our analysis of homomorphisms to reflexive oriented graphs remain the same when we replace oriented graph with 2-edge-coloured graph. In this case, homomorphism requires that existence and colour of edges are preserved. And so we define trivial, improper and proper analogously for homomorphisms 2-edge-coloured graphs.
The main aim of this work is to study the structure of those oriented graphs and 2-edge-coloured graphs that admit no improper homomorphisms. To aid our study we consider the following notion of convexity for oriented and 2-edge-coloured graphs.
Let − → G be an oriented graph and consider S ⊆ V− → G . We say S is convex when no vertex in V− → G \ S is the centre vertex of a 2-dipath whose ends are in S. The convex hull of S, denoted conv(S), is the smallest v u conv (uv) . We analogously define convex and convex hull for 2-edge-coloured graphs by replacing 2-dipath with alternating 2-path in our definition.
Equivalently one may define the convex hull of a set of vertices as follows. Let Γ be an oriented or 2-edge-coloured graph and consider S ⊆ V Γ . Define the sequence S 0 , S 1 , . . . so that where N is the set of vertices that are the centre of a 2-dipath or alternating 2-path whose ends are in S i .
Since S 0 ⊆ S 1 ⊆ . . . and V Γ is finite, there exists a least integer k so that S k = S k+1 . IOne may verify S k = conv(S). This in turn implies that our notion of convex hull is well-defined.
With this definition it is clear that if S ⊆ S, then conv(S ) ⊆ conv(S). By way of example, consider the 2-edge-coloured graph in Figure 1. Smolíková (2002) introduces this notion of convexity for oriented graphs. Duffy and Pas (2018) extend to notion 2-edge-coloured graphs. In both cases the following key observation arises.
Let Γ be an oriented or 2-edge-coloured graph. We say that Γ is complete convex when for each uv ∈ A− → Γ (uv ∈ E Γ ) we have conv(uv) = V Γ . Notice that the 2-edge-coloured graph in Figure 1 is not complete convex. See Figure 2 for an example of a complete-convex oriented graph and complete-convex 2-edge-coloured graph.
Theorem 1.2. Let Γ be an oriented or 2-edge-coloured graph. We have that Γ admits no improper homomorphism if and only if Γ is complete convex. We prove the result for oriented graphs. The result for 2-edge-coloured graphs follows similarly.

Proof: Let
− → G be a complete-convex oriented graph and let − → H be a reflexive oriented graph. If there exists φ :

Let
− → G be an oriented graph so that for every reflexive oriented graph − → H and every homomorphism φ : The mapping c defines a improper homomorphism to a reflexive oriented graph − → H with vertex set {k, k + 1, k + 2, . . . , n} where ij ∈ A H whenever v i v j ∈ A G (k ≤ i < j ≤ n). This contradicts the non-existence of improper homomorphisms of − → G .
Theorem 1.2 implies that we may study oriented or 2-edge-coloured graphs that admit no improper homomorphism using the notion of convexity introduced above. Since one may check in polynomial time whether an oriented or 2-edge-coloured graph is complete convex, we arrive at the following. Theorem 1.3. For any pair of input oriented or 2-edge-coloured graphs Γ 1 and Γ 2 with Γ 2 reflexive, it is Polynomial to decide if the only homomorphisms of Γ 1 to Γ 2 are each either trivial or proper.
The remainder of the work proceeds as follows: In Section 2 we study the structure of oriented and 2-edge-coloured graphs that admit no improper homomorphisms. In doing so we fully classify such oriented and 2-edge-coloured graphs whose underlying graphs have tree-width 2. In Section 3, we show to be NP-complete the problem of deciding if a graph is the underlying graph of some 2-edge-coloured or oriented graph that admits no improper homomorphism. In Section 4 we apply our work on improper homomorphisms to the study of the chromatic number of oriented graphs. In doing so we provide a tool that may aid work on a long-standing open-problem concerning the chromatic number of oriented planar graphs. We discuss this and other impacts of our work in Section 5.

The Structure of Oriented and 2-edge Coloured Graphs that Admit No Improper Homomorphism
We apply Theorem 1.2 and study complete convexity in oriented and 2-edge-coloured graphs. We begin with two näive, but useful, observations. Lemma 2.1. Let − → G and − → H be oriented graphs. If − → G and − → H are each complete convex, then any oriented graph formed by identifying any arc of and (H, c H ) are complete convex, then any 2-edge-coloured graph formed from by identifying a red (blue) edge in (G, c G ) and a red (blue) edge in (H, c H ) is complete convex.
With an eye towards graphs with tree-width 2, we continue our study with an observation of vertices of degree 2 in complete-convex oriented graphs.
Thus s 1 q s 2 is a 2-dipath whose ends are in S and whose centre vertex is in Lemmas 2.3 and 2.4 allow us to fully classify those graphs with tree-width 2 that admit a completeconvex orientation.
Theorem 2.5. Let T be a graph with tree-width 2. An orientation of T , say − → T , is complete convex if and only if T is a 2-tree and every induced copy of If T is a proper subgraph of T , then there is a largest index j such that v j has degree less than 2 in is not complete convex. This is a contradiction. Thus T is a 2-tree.
It remains to show that every induced copy of K 3 is a directed 3-cycle in − → T . Assume otherwise. Thus there exists a greatest index k > 2 so that the copy of Using Theorem 1.2 we re-frame these results to apply to the study of oriented graphs that admit no improper homomorphism.
Corollary 2.6. An orientation of a graph with tree width 2, − → T , admits no improper homomorphism if and only if T is a 2-tree and every induced copy of Lemma 2.4 gives a method to add/remove vertices of degree 2 to a complete-convex oriented graph so that the resulting oriented graph is complete convex: Let − → H be a complete-convex oriented graph with k ≥ 2 vertices. For any arc uw ∈ A− → H one may add a new vertex v and arcs wv, vu to form a completeconvex oriented graph with k + 1 vertices. We now describe a method of adding/removing arcs to a complete-convex oriented graph so that the resulting oriented graph is complete convex.
Let − → G be oriented graph and let a ∈ A− → G . Denote by − → G a the oriented graph formed from − → G by reversing the orientation of a. Denote by −−−→ G − a the oriented graph formed from − → G by removing a.
Theorem 2.8. Let − → G be a complete-convex oriented graph. If uv / ∈ E G and u and v are the ends of a As with Theorem 2.5, each of Theorem 2.7 and 2.8 can be re-interpreted as statements about oriented graphs that admit no improper homomorphism. For 2-edge-coloured graphs a different picture emerges when we examine vertices of minimum degree.
Theorem 2.11. A complete-convex 2-edge-coloured graph is either a monochromatic copy of K 2 or has minimum degree 3.
Proof: Consider G ∼ = K 2 . Let (G, c G ) be a complete-convex 2-edge-coloured graph. Let v be a vertex of minimum degree in G.
If v has a single neighbour, say u, in G, then conv(uv) = {u, v} = V G . Assume v has two neighbours, say x and y, in G. Since (G, c G ) is complete convex, v must be the centre vertex of an alternating 2-path and x and y must be adjacent. Without loss of generality, let c G (xv) = R, c G (vy) = B and c G (xy) = R. However we notice that conv(vy) = {v, y} = V G .
In fact, graphs that are not sufficiently dense cannot be the underlying graph of a complete-convex 2-edge coloured graph.
Theorem 2.12. If G is a graph with at most 2n − 3 edges, then G is not the underlying graph of any complete-convex 2-edge coloured graph.
Proof: Let G be a graph with at most 2n − 3 edges. If G has exactly two vertices, then G is a monochromatic copy of K 2 , which is not complete convex.
Let G be a graph with n ≥ 3 vertices and at most 2n − 3 edges. Let (G, c G ) be a 2-edge-coloured graph. If (G, c G ) is monochromatic then the convex hull of every edge contains only the end points of the edge. Therefore (G, c G ) is not complete convex.
Assume then that (G, c G ) is not monochromatic. Without loss of generality, there are at most n − 2 red edges in (G, c G ). Therefore the spanning subgraph formed by removing all blue edges is not connected.
Let G R be a non-trivial component of this subgraph. Let K b 2 denote the 2-edge-coloured copy of K 2 where the edge between the distinct vertices is blue. Let v 1 and v 2 be the vertices of K b 2 . Consider the homomorphism φ : (G, c G ) → K b 2 given by Since G R necessarily contains at least one edge, this homomorphism is improper. The result now follows by Theorem 1.2.
Corollary 2.13. No 2-degenerate graph is the underlying graph of a complete-convex 2-edge-coloured graph.
Contrasting these results with the statements of Theorem 2.5 and Corollary 2.6, we observe the following: Theorem 2.14. No graph with tree-width 2 is the graph underlying a complete-convex 2-edge-coloured graph.
Corollary 2.15. Every 2-edge-coloured graph whose underlying graph has tree-width 2 admits an improper homomorphism to a monochromatic reflexive copy of K 2 .
Theorems analogous to Theorems 2.7 and 2.8 hold for 2-edge-coloured graphs. For the analogue of Theorem 2.7 the notion of reversing the orientation of an arc is replaced with the notion of changing the colour of an edge. For the analogue of Theorem 2.8 we replace 2-dipath with alternating 2-path.

Complexity of Finding Orientations and 2-edge-colourings that Admit No Improper Homomorphism
In this section we consider the problem of deciding if a graph admits an orientation or a 2-edge-colouring that admits no improper homomorphisms. As with our work in Section 2 we study the problem through the lens of convexity. For this end we define the following decision problems.

Problem: COMPLETE CONVEX 2EC
Instance: A graph G. Decide: Does there exist a complete-convex 2-edge-coloured graph (G, c G )?
Problem: COMPLETE CONVEX ORIENT Instance: A graph G. Decide: Does there exist a complete-convex orientation of G?
Theorems 2.5 and 2.14 imply that there are YES instances of COMPLETE CONVEX ORIENT that are NO instances of COMPLETE CONVEX 2EC. These decision problems are not equivalent. We classify these problems through a reduction from monotone not-all-equal satisfiability.

Problem: MONOTONE NAE3SAT
Instance: A monotone boolean formula Y = (L, F ) in conjunctive normal form with three variables in each clause. Decide: Does there exist a not-all-equal satisfying assignment for the elements of L?
Without loss of generality, we assume that in an instance of MONOTONE NAE3SAT no partition of L induces a partition of F . Theorem 3.1. Schaefer (1978) The decision problem MONOTONE NAE3SAT is NP-complete.
Let Y = (L, F ) be an instance of MONOTONE NAE3SAT. We construct the graph G Y as follows. For each g ∈ F we construct F g as shown in Figure 3. We call these refer to these graphs as clause graphs. Fig. 3: The clause graph, Fg, for g = u ∨ v ∨ w.
We construct G Y from the set of clause graphs for Y as follows: • Identify all vertices labelled b; and • for each x ∈ L identify all vertices labelled x.
Given Y , the graph G Y can be constructed in polynomial time.
We show that there exists complete-convex 2-edge-coloured graph (G Y , c G Y ) if and only if Y is notall-equal satisfiable. For each x ∈ L, the colour of the edge xb will represent the assignment for x in a not-all-equal satisfying assignment for Y . We begin with three technical lemmas.

Lemma 3.2. Consider a complete-convex
Proof: We proceed by contradiction. Without loss of generality, assume c G Two edges of the triangle induced by vertices u g , v g and w g must get the same colour, say c(u g v g ) = c(w g v g ). However now we notice conv{u g w g } = {u g , w g }. This is a contradiction as For a clause f and a variable x that appears as a literal of f , denote by F f,x the subgraph induced by If each of the 2-edge-coloured subgraphs of the form F f,x and C g is complete convex, then (G Y , c G Y ) is complete convex.

Proof:
We proceed by induction on the number of clauses in Y . If Y has a single clause, then G Y = F f , where f is the lone clause in F . By Lemma 2.2, it follows that G Y is complete convex.
Consider now |F | = k > 1 and f ∈ F with f = x ∨ y ∨ z. Without loss of generality, x appears in some other clause of F . (Recall that we may assume that no partition of L induces a partition of F ). Let Y f be the instance of NAE3SAT formed by removing f from F .
For a clause f ∈ F that contains literal x, the colour of the edge x f b will correspond to the value of the literal x in the clause f . Lemma 3.2 implies that for a fixed literal, all such edges have the same colour. Lemma 3.3 implies that for a fixed clause the literals are not all equal. We claim t is not-all-equal satisfying for Y . Consider g ∈ F with g = u∨v∨w. By Lemma 3.3 we have Consider now s : L → {0, 1} so that s is not-all-equal satisfying for Y . We construct c G Y as follows. For all f ∈ F so that x is a literal of f and s(x) = 1 let For all f ∈ F so that x is a literal of f and s( Fig. 4: A 2-edge-coloured clause graph for g = u ∨ v ∨ w, when s(u) = 0 and s(v) = s(w) = 1.
(See Figure 4 for the case s(u) = 0, s(v) = s(w) = 1.) By inspection, each of the 2-edge-coloured subgraphs of the form F f,x and C g is complete convex (See Figures 5 and 6) Thus by Lemma 3.4 we have that (G Y , c G Y ) is complete convex.  Theorem 3.6. The decision problem COMPLETE CONVEX 2EC is NP-complete.
Proof: Verifying that a 2-edge-coloured graph is complete convex is Polynomial. Therefore COMPLETE CONVEX 2EC is in NP. Given Y , the graph G Y can constructed in polynomial time. By Lemma 3.5 and Theorem 3.1, COMPLETE CONVEX 2EC is NP-complete.
Corollary 3.7. It is NP-complete to decide if a graph admits a 2-edge-colouring that admits no improper homomorphism.
We turn now to oriented graphs. We proceed similarly as in the argument for 2-edge-coloured graphs. Given Y = (L, F ) we construct G Y as above. We form G Y from G Y by adding a vertex g and edges gu g , gv g , gw g and removing vertices u g , v g , w g for each clause g = u ∨ v ∨ w. We show there exists a complete convex orientation of G Y if and only if Y is not-all-equal satisfiable. For each x ∈ L the orientation of the edge xb will represent the assignment for x in a not-all-equal satisfying assignment for Y . We begin with three technical lemmas in the spirit of Lemmas 3.2, 3.3 and 3.4. The proofs of these lemmas follow similarly to those of Lemmas 3.2, 3.3 and 3.4 and are thus omitted.
Lemma 3.8. Consider −→ G Y a complete convex orientation of G Y . For every x ∈ L and every f ∈ F so that x is a literal of f we have that x, b, x f is not a 2-dipath.
Lemma 3.9. Consider −→ G Y a complete convex orientation of G Y . For every g = u ∨ v ∨ w we have that b is not a source or sink in the subgraph induced by {u g , v g , w g , b}.
For a clause f and a variable x that appears as a literal of f , denote by F f,x the subgraph induced by {b, x, x f , x f }. For a clause g = u ∨ v ∨ w denote by C g the subgraph induced by {g, b, u g , v g , w g }. Proof: Let Y = (L, F ) be an instance of MONOTONE NAE3SAT.

Consider
We claim t is not-all-equal satisfying for Y . Consider g ∈ F with g = u ∨ v ∨ w. By Lemma 3.9 we have that b is not a source or a sink in the subgraph induced by {u g , v g , w g , b, g}. Therefore For g ∈ F with g = u ∨ v ∨ w so that s(u) = 0, s(v) = s(w) = 1 • orient the edge v g w g to have its tail at w g ; • orient the edge u g g to have its tail at u g ; and • orient u g , v g , g, w g as a directed 4-cycle.
For g ∈ F with g = u ∨ v ∨ w so that s(u) = 1, s(v) = s(w) = 0 • orient the edge v g w g to have its head at w g ; • orient the edge u g g to have its head at u g ; and • orient w g , g, v g , u g as a directed 4-cycle.
(See Figure 7 for the case s(u) = 1, s(v) = s(w) = 0.) For g = u ∨ v ∨ w, we observe that the oriented graph − → C g is complete convex (See Figure 8). For x ∈ L and f ∈ F so that x is a literal of f , we observe that the oriented graph − − → F f,x is complete convex (See Figure 9).
The result now follows from Lemma 3.10. x Theorem 3.12. The decision problem COMPLETE CONVEX ORIENT is NP-complete.
Corollary 3.13. It is NP-complete to decide if a graph admits an orientation that admits no improper homomorphism. (2018) show each of COMPLETE CONVEX EC and COMPLETE CONVEX ORIENT are polynomial when restricted to complete graphs Theorems 2.5 and 2.14 imply that each of COMPLETE CONVEX EC and COMPLETE CONVEX ORIENT are polynomial when restricted to graphs with treewidth 2.

The Chromatic Number of Minor-Closed Families of Oriented Graphs
Recall that one may use homomorphism to define a notion of proper vertex colouring for oriented graph that, in some sense, respects the orientation of the arcs. Let − → G be an irreflexive oriented graph. The chromatic number of − → G is the least integer k that − → G → − → T for some tournament − → T with k vertices. We denote this parameter as χ( − → G ). For a family of irreflexive oriented graphs − → F we define χ( − → F ) to be the least integer k such that χ( F . This notion of chromaticity was first introduced by Courcelle (1994). In this work he shows that formulas expressed in the monadic second-order logic of graph can be decided in polynomial time for graphs with bounded tree-width. A major landmark in the study of such colourings of is the upper bound on the chromatic number of orientations of planar graphs.
Theorem 4.1. Raspaud and Sopena (1994) Let P be an irreflexive planar graph. For any orientation The proof of this result does not rely directly on planarity, rather it follows from the fact that every planar graph admits an acyclic 5-colouring (see Grünbaum (1974).) Thus it remains possible that this bound can be improved. This possibility is buttressed by a complete lack of examples of orientations of planar graphs whose chromatic number is between 19 and 80. Though significant work has gone into the study of the chromatic number of orientations of planar graphs the last 25 years, most meaningful progress has been made on restricted classes of planar graphs, such as those with bounded girth (see Borodin et al. (2007); Borodin and Ivanova (2005); Borodin et al. (1998); Ochem and Pinlou (2014); Pinlou (2009)). A notable exception to this lack of progress for the general case is work by Smolíková (2002) (see Theorem 4.2 below) resulting from the following relaxation of the definition of chromatic number to allow for non-trivial homomorphisms to reflexive targets.
Let − → G be an oriented graph. The simple chromatic number of − → G is the least integer k so that there exists a non-trivial homomorphism − → G → − → T for some reflexive tournament − → T with k vertices. We denote this parameter as χ s ( − → G ). We analogously define χ s ( − → F ).
Theorem 4.2. Smolíková (2002) For − → P the family of orientations of irreflexive planar graphs we have Our work on the study of complete-convex oriented graphs and the definitions above directly imply the following.
With an eye towards the study of orientations of planar graphs, we use this theorem to study the simple chromatic number of minor-closed families of graphs.
Theorem 4.4. Let F be a family of irreflexive graphs that is closed with respect to edge contraction. Let − → F c denote the set of complete-convex orientations of elements of F. We have χ s ( Proof: Let − → F be the set of orientations of some family F that is closed with respect to edge contraction.
Corollary 4.5. If F is a minor-closed family of irreflexive graphs, then χ s ( By way of demonstration, we apply Corollary 4.5 to some well-known families of minor-closed irreflexive graphs.
3. For R the family of irreflexive graphs with tree-width at most 2, we have χ s ( − → R) = 3. Proof: 1. The only elements of − → W c are the directed path with two vertices and the directed cycle with three vertices. The directed 3-cycle has chromatic number 3. By Corollary 4.5 it follows that χ s ( − → W) = 3.
2. The only element of − → G c is the directed path with two vertices. By Corollary 4.5 it follows that χ s ( − → G ) = 2.
3. By Theorem 2.5, the only elements of − → R c are the directed path with two vertices and orientations of 2-trees in which each copy of K 3 is oriented as a directed three cycle. It easily checked that each such orientation of a 2-tree admits a proper homomorphism to the directed three cycle. Thus χ s ( − → R) = 3.
Each of these results appear in previous work (see Duffy and Pas (2018); Smolíková (2002)). However, we note that in each case the result is found by finding a universal target for homomorphism over all graphs in the relevant family. Here this process is simplified by studying only those complete convex elements of the family. As such Corollary 4.5 has the potential to be a powerful tool in studying the simple chromatic number of orientations of minor-closed families. To wit, combining Corollary 4.5 with the statement of Theorem 4.2 yields the following result.
Thus to improve the bound on the chromatic number of orientations of planar graphs given in Theorem 4.1 one may restrict attention to those orientations that are complete convex. We comment further on this approach in Section 5.

Conclusion and Outlook
The similarity in these results for oriented graphs and 2-edge-coloured graphs herein is expected based on previous work on homomorphisms of these objects. Indeed, the respective study of the chromatic number of oriented and 2-edge-coloured graphs is littered with results and techniques that are strikingly similar.
For fixed (m, n) = (0, 0) an (m, n)-mixed graph is a graph that permits m different colours of arcs and n different colours of edges. In particular, graphs are (0, 1)-mixed graphs, oriented graphs are (1, 0)mixed graphs and 2-edge-coloured graphs are (0, 2)-mixed graphs. The definitions of colouring and homomorphism for graphs, oriented graphs and 2-edge-coloured graph can be generalized using (m, n)mixed graphs. It is unsurprising then that definitions, theorems and constructions for oriented and 2-edgecoloured colourings often extend to the more general (m, n)-mixed graph setting. Theorem 4.1 and an analogue of this result for 2-edge-coloured graphs by Alon and Marshall (1998) were shown to be special cases of a more general theorem for (m, n)-mixed graphs by Nešetřil and Raspaud (2000). Theorem 4.2 remains true for planar 2-edge-coloured graphs and (m, n)-mixed graphs (see Duffy (2015); Duffy and Pas (2018)). As do the results in Section 4. Similarities also occur in the study of the chromatic number of bounded degree oriented, 2-edge-coloured and (m, n)-mixed graphs (see Das et al. (2017); Kostochka