All Subgraphs of a Wheel are 5-Coupled-Choosable
AAll Subgraphs of a Wheel are 5-Coupled-Choosable
Sam Barr and Therese BiedlFebruary 8, 2021
Abstract
A wheel graph consists of a cycle along with a center vertex connected to every vertex in the cycle.In this paper show that every subgraph of a wheel graph has list coupled chromatic number at most 5.We further show that this bound is tight for every wheel graph with at least 5 vertices.
In this paper we study the problem of coupled choosability , the problem of finding a valid coloring given listassignments to every vertex and face of a planar graph. The problem is of great relevance to list coloring1-planar graphs, as list coupled coloring a planar graph corresponds to list coloring an optimal 1-planargraph. (Detailed definitions will be given in Section 2.) Wang and Lih [6] show that every planar graph is7-coupled-choosable, and hence every optimal 1-planar graph is 7-choosable. It is an open problem whetherthis bound holds for all 1-planar graphs. They further show that maximal planar graphs are 6-coupled-choosable, planar graphs of maximum degree 3 are 6-coupled-choosable, and both series-parallel graphs andouterplanar graphs are 5-coupled-choosable.It is not hard to characterize k -coupled-choosability for small values of k . Trivially, only the emptygraph is 1-coupled-choosable, and (among the connected graphs) only the single-vertex graph is 2-coupled-choosable. A connected graph is 3-coupled-choosable if and only if it is a tree, for at any edge e incident toa cycle we would require 4 colors for the endpoints of e and the two (distinct) incident faces of e .The above result by Wang and Lih settles the coupled choosability for planar partial 2-trees (which arethe same as the series-parallel graphs). Initially wishing to investigate the coupled choosability of planarpartial 3-trees, in this paper we investigate the coupled choosability of wheel graphs and subgraphs of wheelgraphs. In Theorem 4.3, we show that any subgraph of a wheel is 5-coupled-choosable. Furthermore, inTheorem 4.4, we characterize the coupled choosability of wheel graphs by showing that 5 is tight for wheelgraphs with at least 5 vertices. In the last section of the paper, we touch upon how these results could berelevant in finding the coupled choosability of planar partial 3-trees.As for related results, the (non-coupled) choosability of wheel graphs was characterized in a differentpaper by Wang and Lih [7]: wheels of even order have list chromatic number 4, while wheels of odd orderhave list chromatic number 3. This stands in contrast to our result, as the parity of the number of verticesin the graph does not affect the coupled choosability of wheel graphs. They also show that Halin graphsthat are not wheels have list chromatic number 3, while in Theorem 5.3 we prove the existence of a Halingraph that is not 5-coupled-choosable (in fact, it is not 5-coupled-colorable).Our paper is structured as follows: In Section 2 we will go over the necessary definitions and terminologyfor graphs and graph coloring. In Section 3 we investigate the coupled choosability of wheel graphs. In Section4 we extend this analysis to subgraphs of wheels, along with lower-bounding the coupled choosability of wheelgraphs. In Section 5 we go over several possible extensions to our results and some related open problems. We assume basic familiarity with graph theory (see [2]). In this paper all graphs are finite and connected.We recall that a graph G is called planar if it can be drawn in the plane without edges crossing, and plane if a specific planar drawing Γ is given. The maximal regions of R \ Γ are called faces ; the unbounded1 a r X i v : . [ c s . D M ] F e b egion is known as the outer face and all other faces are inner faces . A bigon is a face that is bounded by twoduplicate edges between a pair of vertices. For a plane graph G , we use V ( G ), E ( G ), and F ( G ) to denotethe set of vertices , the set of edges , and the set of faces of G , respectively. The dual graph G ∗ of a planegraph is obtained by exchanging the roles of vertices and faces, i.e., G ∗ has a vertex for every face of G , andan edge ( f , f ) for every common edge of the two corresponding faces f , f in G .A list assignment is a map L that assigns a set of colors for each vertex or face in V ( G ) ∪ F ( G ). A coupledcoloring with respect to L is a map c such that c ( x ) ∈ L ( x ) for every x ∈ V ( G ) ∪ F ( G ), and L ( x ) (cid:54) = L ( y ) forincident or adjacent elements x, y ∈ V ( G ) ∪ F ( G ). If such a map c exists, then we say that G is L -coupled-choosable . If G is L -coupled-choosable for every L such that | L ( x ) | = k for every x ∈ V ( G ) ∪ F ( G ), thenwe say that G is k -coupled-choosable . The smallest integer k such that such that G is k -coupled-choosable iscalled the list coupled chromatic number of G and denoted χ Lvf ( G ). Observe that a list coupled coloring ofa graph G implies a list coupled coloring of the dual graph G ∗ , since the roles of the vertices and the facesis exchanged but incidences/adjacencies stay the same. Hence, we have χ Lvf ( G ) = χ Lvf ( G ∗ ).A natural way to express the list coupled chromatic number is to define a new graph X ( G ) with verticesfor all vertices and faces of G and edges whenever the vertices and faces G are adjacent/incident. Thisgraph X ( G ) is , i.e., can be drawn in the plane with at most one crossing per edge. In fact, if G is3-connected then X ( G ) is an optimal 1-planar graph , i.e., it is simple and has the maximum-possible 4 n − G correspondsto a vertex coloring of X , i.e., a coloring of the vertices such that adjacent vertices have different colors.When restricting a vertex coloring to given lists L , then the respective terms are L -choosable , k -choosable ,and the list chromatic number χ L ( X ).The wheel graph W n is formed by starting with a cycle C n − on n − outer cycle ), addinga center vertex inside the cycle and adding an edge from the center vertex to every vertex on the cycle. Wewill label the center vertex and the outer face of the wheel graph as x and f , respectively. We furtherlabel the vertices in the outer cycle as x , . . . , x n − , and label the inner faces as f , . . . , f n − such that x i isincident to f i and f i +1 for 1 ≤ i < n −
1, and x n − is adjacent to f n − and f (see Figure 1).An outerplanar graph is a graph that can be drawn in the plane such that every vertex is on the outerface. A subdivision of a graph G is formed by repeatedly taking some edge uv ∈ E ( G ), removing e from G ,adding a new vertex x , and adding edges ux and xv . In order to prove the desired result for all subgraphs of the wheel graph, we first determine the coupledchoosability of the wheel graph itself. It will be helpful to recall the following result relating the choosabilityof a graph to the maximum degree; it is an analogue to Brook’s theorem and similarly upper-bounds thechromatic number of a graph by its maximum degree.
Lemma 3.1. (Erd˝os, Rubin, and Taylor [3]) Let G be a connected graph that is neither an odd cycle nor acomplete graph. Then G is ∆( G ) -choosable. Our main result in this section is:
Lemma 3.2.
Every wheel graph W n , n ≥ , is 5-coupled-choosable.Proof. For n = 4, W is the complete graph K . Wang and Lih [6] proved that χ Lvf ( K ) = 4, so we assume n ≥
5. Let L be a color assignment for W n such that | L ( y ) | = 5 for every y ∈ V ( W n ) ∪ F ( W n ). Our goal isto find a coupled coloring with respect to L . Since x and f are both adjacent to all remaining vertices, wewill color them first and then color the rest. This rest defines the following graph X n : • The vertices of X n correspond to vertices x , . . . , x n − and faces f , . . . , f n − of W n . • There is an edge x i x j if the vertices x i and x j are adjacent in W n . • There is an edge f i f j if the faces f i and f j are adjacent in W n . • There is an edge x i f j if the vertex x i is incident to the face f j in W n .2bserve that | V ( X n ) | = 2 n − X n is 4-regular (see Figure 1). Furthermore, it suffices to find avertex-colouring of X n with respect to L , plus two suitable colors in L ( x ) and L ( f ) for x and f . We havetwo cases: x x x x n − x n − x f f f f n − f n − f x f x f x x n − f n − x n − f n −
433 44 33 3 3 33 33 334
Figure 1: The graph W (left) and X (right). Circled numbers indicate a lower bound on the list-length in L (cid:48) . Case 1: L ( x ) ∩ L ( f ) (cid:54) = ∅ . Let a ∈ L ( x ) ∩ L ( f ), and assign a to x and f . Observe that | L ( y ) \ { a }| ≥ y ∈ V ( X n ) and X n has maximum degree 4. Moreover | X n | = 2 n − X n is not an oddcycle. Also x and x are not adjacent by n ≥
5, so X n is not a complete graph. Therefore, by Lemma3.1, we have a list coloring of the vertices of X n that only uses colors in L \ { a } , which in turn implies an L -list-coloring of the vertices and faces of W n . Case 2: L ( x ) ∩ L ( f ) = ∅ . We find suitable colors for x and f by imitating the method used for K in [6](but adapted here to 5 colors). Define color-pairs S := {{ a, b } : a ∈ L ( x ) , b ∈ L ( f ) } . By case-assumption | S | = 25.We claim that | { s ∈ S : s ⊆ L ( y ) } | ≤ y ∈ V ( X n ). To see this, let y ∈ V ( X n ), and consider thedisjoint sub-lists L := L ( y ) ∩ L ( x ) and L := L ( y ) ∩ L ( f ). Since | L | + | L | ≤ | L ( y ) | = 5, and | L | and | L | are integers, we have | { s ∈ S : s ⊆ L ( y ) } | = | L × L | = | L | · | L | ≤ . Therefore, color-pairs of S appear as subsets of lists in X n at most (cid:88) y ∈ X n | { s ∈ S : s ⊆ L ( y ) } | ≤ (2 n − · n − | S | = 25, some element { a (cid:48) , b (cid:48) } of S appears at most12 n − < n − X n . Color x with a (cid:48) and f with b (cid:48) . For y ∈ V ( X n ), define L (cid:48) ( y ) := L ( y ) \{ a (cid:48) , b (cid:48) } .For any y ∈ V ( X n ), we have 3 ≤ | L (cid:48) ( y ) | ≤
5. We call y a if | L (cid:48) ( y ) | = 3 (this implies { a (cid:48) , b (cid:48) } ⊂ L ( y )),and a otherwise. From our choice of colors a (cid:48) and b (cid:48) , we have | { y ∈ V ( X n ) : y is a 3-vertex } || V ( X n ) | < ( n − / n − X n are 4-vertices. Consider the cyclic enumeration σ := (cid:104) f , x , f , x , . . . , f n − , x n − (cid:105) of the vertices of X n . Since strictly more than | V ( X n ) | of the vertices are 4-vertices, we have four consecutive4-vertices in σ . Up to exchange of f i and x i and renumbering, we may assume that f , x , f , and x are4-vertices. Figure 1(right) illustrates the lower bounds on the size of L (cid:48) .3e next color f n − , x n − and x and have two sub-cases. If L (cid:48) ( f n − ) ∩ L (cid:48) ( x ) (cid:54) = ∅ , then color f n − and x with the same color. Otherwise, since | L (cid:48) ( f n − ) ∪ L (cid:48) ( x ) | ≥ > | L ( f ) | , there are colors p and q for f n − and x respectively such that at least one of them is not in L ( f ), i.e., | L ( f ) ∩ { p, q }| ≤
1. Pick these colorsfor f n − and x . In either case, two vertices adjacent to x n − have been colored, and | L (cid:48) ( x n − ) | ≥
3, so x n − will have at least one valid color left, and we pick this color for x n − .We now have colors p, q , and r for f n − , x , and x n − (respectively) such that | L (cid:48) ( f ) ∩ { p, q, r }| ≤ L (cid:48)(cid:48) such that | L (cid:48)(cid:48) ( f ) | = | L (cid:48) ( f ) \ { p, q, r }| ≥ − | L (cid:48)(cid:48) ( f ) | = | L (cid:48) ( f ) \ { q }| ≥ − | L (cid:48)(cid:48) ( x ) | = | L (cid:48) ( x ) \ { q }| ≥ − | L (cid:48)(cid:48) ( x n − ) | = | L (cid:48) ( x n − ) \ { p, r }| ≥ − | L (cid:48)(cid:48) ( f n − ) | = | L (cid:48) ( f n − ) \ { p }| ≥ − | L (cid:48)(cid:48) ( x i ) | ≥ ≤ i ≤ n − | L (cid:48)(cid:48) ( f i ) | ≥ ≤ i ≤ n − f f x f x x n − f n − pqr Figure 2: The graph X (cid:48) n (solid). Dotted edges show the rest of the graph X n , along with a dashed edge tothe vertex f which remains to be colored. Numbers indicate lower bounds on the list length in L (cid:48)(cid:48) .Let X (cid:48) n := X n \ { f n − , x n − , f , x } and color it with respect to list assignment L (cid:48)(cid:48) . This is feasible since X (cid:48) n is outerplanar (see Figure 2), and outerplanar graphs are 3-choosable even if the colors of two consecutivevertices on the outer face are fixed [4] (here we fix the colors for x n − and f n − ). This colors all verticesexcept for f , but | L (cid:48)(cid:48) ( f ) | ≥ f has only one neighbor in X (cid:48) n , so we can give it a color not used by f .Therefore, we have a list vertex-coloring of X n that is compatible with the colors for x , f chosen earlierand so implies a list coupled coloring of W n . Now we turn to graphs that are subgraphs of wheels. In contrast to list coloring the vertices of the graph,there is no clear relationship between the list coupled chromatic number of a graph and the list coupledchromatic number of its subgraphs. Indeed, it is possible for the subgraph to have a larger list coupledchromatic number.
Observation 1.
There exists a plane graph G with subgraph H ⊆ G such that χ Lvf ( H ) > χ Lvf ( G ) Proof.
Consider the graph K , and the subgraph H obtained by deleting one edge; see Figure 3. FromTheorem 10 of [6], we know that the graph K is 4-coupled-choosable, i.e χ Lvf ( K ) = 4. But in graph H ,observe that the incidences and adjacencies between x , x , x , f , and f (cid:48) form a K , and therefore χ Lvf ( H ) ≥ > χ Lvf ( K )4 x x x f f f f x x x x f f f (cid:48) Figure 3: The graph K (left) and subgraph H (right)(Observe that χ Lvf ( H ) = 5 since it is outerplanar.)Therefore non-trivial work is required to demonstrate that any subgraph of a wheel graph is also 5-coupled-choosable. In proving this, it will be helpful to know how the coupled choosability of a graph G relates to the coupled choosability of subdivisions of G . Lemma 4.1.
For any plane graph G , any subdivision H of G is max { , χ Lvf ( G ) } -coupled-choosable.Proof. Let L be a list assignment for H such that | L ( x ) | = max { , χ Lvf ( G ) } for every vertex and face of H .We prove the statement by induction on the number of subdivisions performed on G to obtain H . If H isthe result of subdividing the edges of G zero times, then H = G and so trivially any L -coupled-coloring of G is an L -coupled-coloring of H .Otherwise, H was the result of performing k + 1 subdivisions on G for some k ≥
0. In particular, H is theresult of subdividing a single edge of some graph H (cid:48) , where H (cid:48) was the result of performing k subdivisionson G . Let uv ∈ E ( H (cid:48) ) be the edge of H (cid:48) that was subdivided, and let x be the vertex which was added.By the inductive hypothesis, H (cid:48) is max { , χ Lvf ( G ) } -coupled-choosable. Color the faces of H and the vertices V ( H ) \ { x } according to how they would be colored in H (cid:48) . Then we only need to color the remaining vertex x . Note that x has degree two with neighbors u and v . Let f and f be the two faces adjacent to the edge uv in H (cid:48) . Then u, v, f , and f are the only vertices and faces that are adjacent (respectively incident) to x .Hence, after coloring the vertices and faces from H (cid:48) , x still has at least | L ( x ) | − ≥ − x can always be colored.This implies another results that we will not need but find interesting. For a planar graph G , subdividingan edge corresponds in the dual graph G ∗ to duplicating edges to form bigons. Since χ Lvf ( G ) = χ Lvf ( G ∗ ) wetherefore have: Corollary 4.2.
Let G be a plane graph, and H the result of duplicating some edges of G to form bigons. If χ Lvf ( G ) ≥ , then H is χ Lvf ( G ) -coupled-choosable. If χ Lvf ( G ) ≤ , then H is 5-coupled-choosable. Now we can show our main result.
Theorem 4.3.
Let G be a subgraph of a wheel graph W n , n ≥ . Then G is 5-coupled-choosable.Proof. We examine several possibilities of the structure of G . Case 1: G = W n . Then by Lemma 3.2 G is 5-coupled-choosable. Case 2: G is the result of deleting at least one edge or vertex of W n that is on the outer face. Then G isouterplanar, and so by Theorem 14 in [6] G is 5-coupled-choosable. Case 3: G is the result of removing the center vertex of W n . Then G = C n − has only two faces. Colorthem arbitrarily, which leaves at least three colors for every vertex of C n − . But C n − is outer-planar andhence 3-choosable. Case 4:
None of the above. Then all vertices of W n belong to G , but we deleted some edges which were noton the outer face. So G is the result of deleting some of edges incident to the center vertex ( spoke edges).If at most one spoke remains, then G is outer-planar (after inverting the drawing so that the face incident5o the center vertex becomes the outer-face), and therefore G is 5-coupled choosable. If exactly two spokesremain, then G is a subdivision of a triple edge. A triple edge has two vertices and three faces and thereforeis 5-coupled choosable; by Lemma 4.1 so is G . Finally if at least three spokes remain, then G is a subdivisionof some W k for k ≥
4, and by Lemmas 3.2 and 4.1 G is 5-coupled-choosable.Following the steps of our proof, one can easily verify that we can find the L -coupled-coloring in lineartime. Finding the coloring from Lemma 3.1 is known to be linear time [5].Having established an upper bound on the list coupled chromatic number of wheel graphs in Lemma3.2, one might wonder whether this bound is tight or not. In [6], it is shown that the graph K = W is4-coupled-choosable. In fact, this is the only wheel graph which is 4-coupled-choosable. For all other wheelgraphs, the bound of 5-coupled-choosability is tight. Theorem 4.4. χ Lvf ( W n ) = 5 , for n ≥ .Proof. From Lemma 3.2, we know that wheel graphs are always 5-coupled-choosable. It remains to showthat they are not 4-coupled-choosable for n ≥ n = 5 ,
6, we consider the list assignment L such that L ( y ) = { , , , } for every y ∈ V ( W n ) ∪ F ( W n ).(So these graphs are not even 4-coupled-colorable.) Assume for contradiction that we have an L -coupled-coloring c of W n . If c ( x ) (cid:54) = c ( f ), then this leaves two colors for coloring the triangle x , f , f in X n ,impossible. Hence c ( x ) = c ( f ), say they are both colored 4. Then we have an L (cid:48) -coloring of X n with lists L (cid:48) ( y ) := L ( y ) \ { } = { , , } . Observe that for X and X , any putative L (cid:48) -coloring would be unique up to renaming the colors, sinceonce we have colored one triangle, every other vertex can be reached via a sequence of triangles. One verifiesthat for these graphs (and indeed every X k where k − n = 5 , n ≥
7, we construct a list assignment L such that W n is not L -coupled-choosable. Set L ( x ) = { , , , } and L ( f ) = { , , , } . We further define: L ( f ) = L ( x ) = L ( f ) = { , , , } L ( x ) = L ( f ) = L ( x ) = { , , , } L ( f ) = L ( x ) = L ( f ) = { , , , } L ( x ) = L ( f ) = L ( x ) = { , , , } Observe that each of these triples forms a triangle in X n , and for any a ∈ { , , , } and b ∈ { , , , } ,one of these triangles has colors { a, b, x, y } for some colors x, y . Assume for contradiction that we have an L -coupled-coloring c of W n . Up to symmetry, assume c ( x ) = 1 and c ( f ) = 5. But then f , x , and f havetwo colors left, and therefore cannot be colored, a contradiction.Figure 4: The graphs X (left) and X (right).With this, we have a characterization of the coupled choosability of wheel graphs. Corollary 4.5.
For a wheel graph W n , we have χ Lvf ( W n ) = (cid:40) n = 45 n ≥ Future Work
Our investigation of wheel graphs was motivated by wanting to determine the coupled choosability numberof planar partial 3-trees. To define these, we first define
Apollonian networks recusively as follows. A triangleis an Apollonian network. If G is an Apollonian network, and f is a face of G (necessarily a triangle) thatis not the outer-face, then the graph obtained by stellating face f is also an Apollonian network. Here stellating means the operation of inserting a new vertex v inside face f and making it adjacent to all verticesof f . A planar partial 3-tree is a graph that is a subgraph of an Apollonian network (see Figure 5). (Thisdefinition is different, but equivalent, to the “standard” definition of partial 3-trees via treewidth or viachordal supergraphs with clique-size 4 [1].) We offer the following conjecture: Conjecture 5.1.
Every planar partial 3-tree is 6-coupled-choosable.
Figure 5: A planar partial 3-tree. Dotted edges show the Apollonian network.Note that the conjecture holds for Apollonian networks, since these are maximal planar graphs and theseare known to be 6-coupled-choosable (Wang and Lih [6]). But this does not imply 6-coupled-choosability ofsubgraphs, and so the conjecture remains open.Towards the conjecture, we studied several graph classes that are planar partial 3-trees (and generalizewheels). One such class of graphs are the
Halin graphs , which are defined by starting with a tree T andadding a cycle between the leaves of T . See also the solid edges in Figure 6. Wheel graphs are the specialcase of Halin graphs where T is a star graph. A second class of planar partial 3-trees are the stellatedouter-planar graphs , obtained by starting with some outerplanar graph G , and stellating the outer-face. Seealso the dashed edges in Figure 6. Wheel graphs are the special case of stellated outerplanar graphs wherethe outerplanar graph is a cycle. These two classes are closely related. Lemma 5.2.
For n ≥ , Halin Graphs are exactly the duals of stellated outerplanar graphs. We suspect that this result was known before, but have not been able to find a reference and thereforeprovide a proof here.
Proof.
Let G be a Halin graph. Every face of G that is not the outer face is adjacent to the outer face.Therefore, the weak dual (i.e., the dual graph with the vertex representing the outer face removed) of G isan outerplanar graph. Then adding the outer face and its adjacencies creates a stellated outerplanar graph.Let G be a stellated outerplanar graph. The faces of the outerplanar graph form a tree T in the dual. Thefaces incident to the vertex that stellated the outer face form a cycle L , and every such face of L shares anedge with some face in T . Hence we can view L as a set of leaves attached to T and then further connectedwith a cycle, so this is a Halin-graph.Therefore, any list coupled coloring of a stellated outerplanar graph corresponds to a list coupled coloringof a Halin graph. Unfortunately, our upper bound for the coupled choosability of wheel graphs does not ingeneral extend to Halin graphs. Theorem 5.3.
There exists a stellated outerplanar graph (equivalently a Halin graph) that is not 5-coupled-choosable. roof. The Halin-graph G is the triangular prism, see Figure 6 where we also show the dual graph G ∗ andthe 1-planar graph X ( G ). We claim that X ( G ) is not 5-colorable; therefore G is not 5-coupled-colorable andin particular not 5-coupled-choosable.Assume for contradiction that X ( G ) had a 5-coloring; up to symmetry we may assume that the triangleformed by the three degree-4-faces of G is colored 1 , ,
3. Let ( t, t (cid:48) ) be the edge that crosses the edge coloredwith 2 and 3. Vertices t, t (cid:48) are colored with 1, 4 or 5; up to renaming of colors 4 and 5 hence one of them iscolored 4.Starting with this coloring, propagate restrictions on the possible colors to other vertices of X along thenumerous copies of K (note that all vertices other than t, t (cid:48) are adjacent to the one colored 1). This leadsto a triangle that has only two possible colors left, a contradiction.
12 53 4 53524,54,54,5 t t (cid:48)
Figure 6: A Halin-graph G (black solid; the tree is bold), and the dual graph G ∗ (blue dashed) which isa stellated outerplanar graph (the outerplanar graph is bold). Taking both, and adding the face-vertexincidences (red dotted) gives graph X ( G ). We also show the only possible 5-coloring (up to symmetry) of X ( G ), which leads to a contradiction since a triangle must be colored with 2 colors.In particular, this shows that we cannot replace ‘6’ by ‘5’ in Conjecture 5.1.So wheels are strictly better (as far as coupled choosability is concerned) than Halin-graphs. Now westudy a second graph class that lies between the wheels and the planar partial 3-trees. These are the IO-graphs , which are the planar graphs that can be obtained by adding an independent set to the interior facesof an outerplanar graph (see Figure 7). Certainly any subgraph of a wheel is an IO graph.
Conjecture 5.4.
Every IO-graph is 5-coupled choosable.
We studied subgraphs of wheel graphs because they may be an important stepping stone towards Conjec-ture 5.4. In particular, consider some IO-graph G . obtained from an outerplanar graph O and independentset I . Let G + be a maximal IO-graph containing G , i.e., add edges to G for as long as the result is simple andan IO-graph. Then G + is a tree of wheels, where each wheel consists of a vertex x ∈ I with its neighbours,and the wheels have been glued together at edges. Correspondingly G is a tree of subgraphs of wheels. Itmay be possible to use Theorem 4.3 (enhanced with further restrictions on the coloring of some parts) toprove Conjecture 5.4 by building a coloring of G incrementally in this tree, but this remains future work. References [1] T. Biedl and L.E. Ruiz Vel´azquez. “Drawing planar 3-trees with given face areas”. In:
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Graph theory . Fifth. Vol. 173. Graduate Texts in Mathematics. Springer, Berlin, 2018,pp. xviii+428. isbn : 978-3-662-57560-4; 978-3-662-53621-6.[3] Paul Erd˝os, Arthur L. Rubin, and Herbert Taylor. “Choosability in graphs”. In:
Proceedings of the WestCoast Conference on Combinatorics, Graph Theory and Computing (Humboldt State Univ., Arcata,Calif., 1979) . Congress. Numer., XXVI. Utilitas Math., Winnipeg, Man., 1980, pp. 125–157.8igure 7: An IO graph G consists of an outerplanar graph (circles) and an independent set (squares). Dottededges at added to obtain G + , and some of the wheels used to build G + are shaded.[4] Joan P. Hutchinson. “On list-coloring extendable outerplanar graphs”. In: Ars Math. Contemp. issn : 1855-3966. doi : . url : https://doi.org/10.26493/1855-3974.179.189 .[5] San Skulrattanakulchai. “∆-List vertex coloring in linear time”. In: Information Processing Letters issn : 0020-0190. doi : https://doi.org/10.1016/j.ipl.2005.12.007 . url : .[6] Weifan Wang and Ko-Wei Lih. “Coupled choosability of plane graphs”. In: J. Graph Theory issn : 0364-9024. doi : . url : https://doi.org/10.1002/jgt.20292 .[7] Weifan Wang and Ko-Wei Lih. “List coloring Halin graphs”. In: Ars Combin.
77 (2005), pp. 53–63. issnissn