Preimages of Small Geometric Cycles

A graph G is a homomorphic preimage of another graph H, or equivalently G is H-colorable, if there exists a graph homomorphism f : G → H. A classic problem is to characterize the family of homomorphic preimages of a given graph H. A geometric graph G is a simple graph G together with a straight line drawing of G in the plane with the vertices in general position A geometric homomorphism (resp. isomorphism) G → H is a graph homomorphism (resp. isomorphism) that preserves edge crossings (resp. and noncrossings). The homomorphism poset G of a graph G is the set of isomorphism classes of geometric realizations of G partially ordered by the existence of injective geometric homomorphisms. A geometric graph G is H-colorable if G→ H for some H ∈ H. In this paper, we provide necessary and sufficient conditions for G to be Cn-colorable for 3 ≤ n ≤ 5. 1 Basic Definitions A graph homomorphism f : G → H is a vertex function such that for all u, v ∈ V (G), uv ∈ E(G) implies f(u)f(v) ∈ E(H). If such a function exists, we write G → H and say that G is homomorphic to H, or equivalently, that G is a homomorphic preimage of H. A proper n-coloring of a graph G is a homomorphism G→ Kn; thus, G is n-colorable if and only if G is a homomorphic preimage of Kn. (For an excellent overview of the theory of graph homomorphisms, see [4].) In 1981, Maurer, Salomaa and Wood [9] generalized this notion by defining G to be H-colorable if and only if G → H. They used the notation L(H) to denote the family of H-colorable graphs. For example, G is C5colorable if and only if G→ C5; this means there exists a proper 5-coloring of G such a vertex of color 1 can only be adjacent to vertices of color 2 or 5, but not to vertices of color 3 or 4, etc. Maurer et al. noted that for odd m and n, Cm is Cn-colorable (i.e. Cm → Cn) if and only if m ≥ n. Since any composition of graph homomorphisms is also a graph homomorphism, this generates the following hierarchy among color families of cliques and odd cycles. . . .L(C2n+1) ( L(C2n−1) ( · · · ( L(C5) ( L(C3) = = L(K3) ( L(K4) ( · · · ( L(Kn) ( L(Kn+1) . . . For a given graph H, the H-coloring problem is the decision problem, “Is a given graph H-colorable?” In 1990, Hell and Nes̆etr̆il showed that if χ(H) ≤ 2, then this problem is polynomial and if χ(H) ≥ 3, then it is NP-complete [3]. The concept of H-colorability can been extended to directed graphs. Work has been done by Hell, Zhu and Zhou in characterizing homomorphic preimages of certain families of directed graphs, including oriented cycles [12], [8], [5], oriented paths [7] and local acyclic tournaments [6]. In [1], Boutin and Cockburn generalized the notion of graph homomorphisms to geometric graphs. A geometric graph G is a simple graph G together with a straight-line drawing of G in the plane with vertices in general position (no three vertices are collinear and no three edges cross at a single point). A geometric graph G with underlying abstract graph G is called a geometric realization of G. The definition below formalizes what it means for two geometric realizations of G to be considered the same. Definition 1.1. A geometric isomorphism, denoted f : G→ H, is a function f : V (G)→ V (H) such that for all u, v, x, y ∈ V (G), 1. uv ∈ E(G) if and only if f(u)f(v) ∈ E(H), and 2. xy crosses uv in G if and only if f(x)f(y) crosses f(u)f(v) in H. Relaxing the biconditionals to implications yields the following. Definition 1.2. A geometric homomorphism, denoted f : G → H, is a function f : V (G)→ V (H) such that for all u, v, x, y ∈ V (G), 1. if uv ∈ E(G), then f(u)f(v) ∈ E(H), and 2. if xy crosses uv in G, then f(x)f(y) crosses f(u)f(v) in H. If such a function exists, we write G→ H and say that G is homomorphic to H, or equivalently that G is a homomorphic preimage of H. An easy consequence of this definition is that no two vertices that are adjacent or co-crossing (i.e. incident to distinct edges that cross each other) can have the same image (equivalently, can be identified) under a geometric homomorphism. Boutin and Cockburn define G to be n-geocolorable if G → Kn, where Kn is some geometric realization of the n-clique. The geochromatic number of G, denoted X(G), is the smallest n such that G is n-geocolorable. Observe that if a geometric graph of order n has the property that no two of its vertices can be identified under any geometric homomorphism, then X(G) = n. The existence of multiple geometric realizations of the n-clique for n > 3 necessarily complicates the definition of geocolorability, but there is additional structure we can take advantage of. Definition 1.3. Let G and Ĝ be geometric realizations of G. Then set G Ĝ if there exists a (vertex) injective geometric homomorphism f : G→ Ĝ. The set of isomorphism classes of geometric realizations of G under this partial order, denoted G, is called the homomorphism poset of G. Hence, G is n-geocolorable if G is homomorphic to some element of the homomorphism poset Kn. In [2], it is shown that K3,K4 and K5 are all chains. Hence, for 3 ≤ n ≤ 5, G is n-geocolorable if and only if G → Kn, where Kn is the last element of the chain. By contrast, K6 has three maximal elements, so G is 6-geocolorable if and only if it is homomorphic to one of these three realizations. Definition 1.4. Let H denote the homomorphism poset of geometric realizations of a simple graph H. Then G is H-geocolorable if and only if G→ H for some maximal H ∈ H. In this paper, we provide necessary and sufficient conditions for G to be Cn-geocolorable, where 3 ≤ n ≤ 5. The structure of the homomorphism posets Cn for 3 ≤ n ≤ 5 is given in [2]. It is worth noting that the geometric cycles are richer than than abstract cycles. All even cycles are homomorphically equivalent to K2, and as noted earlier, C2k+1 → C2`+1 if and only if k ≥ `. However, since geometric homomorphisms preserve edge crossings, and both K2 and C3 have only plane realizations, this is not true even for small non-plane geometric cycles, as shown in Figure 1. Figure 1: Ĉ4 6→ K2 and Ĉ5 6→ C3 2 Edge-Crossing Graph and Thickness Edge Colorings Definition 2.1. [2] The edge-crossing graph of a geometric graph G, denoted by EX(G), is the abstract graph whose vertices correspond to the edges of G, with adjacency when the corresponding edges of G cross. Clearly, non-crossing edges ofG correspond to isolated vertices of EX(G). In particular, G is plane if and only if EX(G)→ K1. To focus on the crossing structure of G, we let G× denote the geometric subgraph of G induced by its crossing edges. Note that EX(G×) is simply EX(G) with any isolated vertices removed. From [2], a geometric homomorphism G → H induces a geometric homomorphism G× → H× as well as graph homomorphisms G→ H and EX(G)→ EX(H). Definition 2.2. [1] A thickness edge m-coloring of a geometric graph G is a coloring of the edges of G with m colors such that no two edges of the same color cross. The thickness of G is the minimum number of colors required for a thickness edge coloring of G. From these two definitions, a thickness edge m-coloring of G is a graph homomorphism : EX(G)→ Km. This can be generalized as follows. Definition 2.3. A thickness edge Cm-coloring on G is a graph homomorphism : EX(G)→ Cm. Observe that under a thickness edge Cm-coloring, edges are colored with colors numbered 1, 2, . . . ,m such that colors assigned to edges that cross each other must be consecutive mod m. Equivalently, edges of color i may only be crossed by edges of colors i − 1 and i + 1 mod m. Note also that if G has a thickness edge Cm-coloring for m > 3, then G cannot have three mutually crossing edges. Definition 2.4. Let be a thickness edge coloring G. The plane subgraph of G induced by all edges of a given color is called a monochromatic subgraph of G under . The monochromatic subgraph corresponding to edge color i is called the i-subgraph of G under . We assume from now on that G has no isolated vertices, which implies that every vertex belongs to at least one monochromatic subgraph of G under any thickness edge coloring. 3 Easy Cases: n = 3 and n = 4 The smallest (simple) cycle is C3 = K3. As noted in [1], G → K3 if and only if G is a 3-colorable plane geometric graph. Thus G is C3-geocolorable if and only if G is 3-colorable and EX(G) is 1-colorable, or more concisely, G→ C3 ⇐⇒ G→ K3 and EX(G)→ K1. Next, C4 has two geometric realizations, one plane and the other with a single crossing, which we denote C4 and Ĉ4 respectively. Since C4 → Ĉ4, the homomorphism poset C4 consists of a two element chain, as shown in Figure 2. Hence G is C4-geocolorable if and only if G→ Ĉ4.