on list-coloring outerplanar graphs

On List-ColoringOuterplanar Graphs

Joan P. Hutchinson

MACALESTER COLLEGESAINT PAUL, MINNESOTA 55105

E-mail: [email protected]

Received April 6, 2007; Revised February 5, 2008

Published online 18 June 2008 in Wiley InterScience(www.interscience.wiley.com).DOI 10.1002/jgt.20326

Abstract: We prove that a 2-connected, outerplanar bipartite graph (re-spectively, outerplanar near-triangulation) with a list of colors L(v) for eachvertex v such that |L(v)| ≥ min{deg(v), 4} (resp., |L(v)| ≥ min{deg(v), 5}) canbe L-list-colored (except when the graph is K3 with identical 2-lists). Theseresults are best possible for each condition in the hypotheses and bounds.© 2008 Wiley Periodicals, Inc. J Graph Theory 59: 59–74, 2008

Keywords: list coloring; outerplanar graphs; Gallai tree

1. INTRODUCTION

We study the question of list-coloring outerplanar graphs, in general and bipartitein particular. These questions arise in the study of coloring extension problems[1–3,9,14] in which part of a graph is precolored and a coloring-extension to theentire graph is sought. The extension problem leads naturally to list-coloring sinceprecolored vertices force a reduced list of colors that are allowable on neighboringvertices.

It is well-understood when a graph can be list-colored if each vertex has a listof size equal to its degree. In [4,5,7,17,19] it is shown that such a graph can belist-colored if and only if the graph is not a Gallai tree, that is, a connected graphwhose blocks are either odd cycles or complete graphs; see [12] for a history anda generalization to hypergraphs. Also an outerplanar graph is easily seen to be

Journal of Graph Theory© 2008 Wiley Periodicals, Inc.

59

60 JOURNAL OF GRAPH THEORY

3-choosable since it contains vertices of degree 2. The question we ask involvesa mixture of these two hypotheses: if a 2-connected outerplanar graph satisfies|L(v)| ≥ min{deg(v), 4} for every vertex v, is it L-list colorable provided it doesnot form a Gallai tree of maximum degree at most 4? Theorem 1 shows that theanswer is yes when the graph is bipartite, but the answer is no in general due toan example of Kostochka [11]; see Figure 3. The answer is also no for outerplanarbipartite graphs with a separating vertex, for 2-connected, bipartite graphs that areK4-minor-free, and for 2-connected, outerplanar bipartite graphs when the lowerbound of 4 is replaced by 3.

Theorem 3 shows that if the graph is a 2-connected, outerplanar near-triangulation (or equivalently, edge-maximal outerplanar) and satisfies |L(v)| ≥min{deg(v), 5} for every vertex v, then the graph is L-list-colorable except for K3

with three identical 2-lists. Connectivity cannot be reduced to allow separating ver-tices, and the result does not hold for 2-connected K4-minor-free graphs. It maybe that Theorem 3 holds with only the hypotheses of being 2-connected and outer-planar. In both settings, bipartite and near-triangulation, it is easy to find examplesshowing that deg(v) cannot be replaced by deg(v) − 1.

2. BACKGROUND

We follow the definitions of [20] and consider only finite, simple graphs. A graphhas a (vertex) coloring with k colors if one of k colors is (or can be) assigned toeach vertex so that adjacent vertices receive different colors. Then the graph is saidto be k-colored (or k-colorable). The minimum value of k for which a graph has ak-coloring is known as its chromatic number. Independently in [19,7] the idea of“list-coloring” was introduced and developed. Suppose each vertex v of a graph Gis assigned a list of colors L(v). Then G is said to be L-list-colorable when it has acoloring in which each vertex v receives a color from its list L(v). If each list of Lhas at least k colors and G is L-list-colorable, then G is said to be k-list-colorableor k-choosable, and the minimum k for which G is k-choosable is called its choicenumber.

A connected graph has a separating vertex if there is a vertex whose removalleaves a disconnected graph; a connected graph is 2-connected if it contains noseparating vertex. A block of a graph is a maximal 2-connected subgraph.

In this article we consider classes of simple plane graphs, that is, simple graphsthat are drawn in the plane with no edge crossing. A face of a plane graph is amaximal connected component of the plane minus the vertices and edges of thegraph. In a plane graph there is precisely one face that is unbounded; we callthis the infinite or external face, and all others are finite faces. An edge is calledexternal (respectively, internal) when it lies (resp., does not lie) on the boundary ofthe external face. A face is called a triangle (respectively, quadrilateral, k-sided fork > 2) when it has three (resp., four, k) edges on its boundary. A near-triangulation(respectively, near-quadrangulation) is a plane graph in which all finite faces are

Journal of Graph Theory DOI 10.1002/jgt

ON LIST-COLORING OUTERPLANAR GRAPHS 61

triangles (resp., quadrilaterals). Let Kn denote the complete graph on n vertices andKm,n the complete bipartite graph on m + n vertices, m, n > 0. A graph is said tobe Kn-minor-free if it cannot be transformed into Kn by deleting vertices and edgesand by contracting pairs of adjacent vertices, each pair forming a new vertex. It iswell-known (e.g., see [6]) that K4-minor-free graphs are the subgraphs of graphsobtained by recursively pasting triangles together along edges. (The 2-connectedK4-minor-free graphs are also called series-parallel.) A graph is called outerplanarif all vertices lie on the external face, and a finite face of an outerplanar graph iscalled internal if it has only internal edges on its boundary. Thus outerplanar graphare K4-minor-free, and edge-maximal outerplanar graphs are 2-connected near-triangulations.

The weak dual G∗w of a plane graph G has a vertex for each finite face (but not

for the infinite face), with two vertices adjacent whenever the corresponding facesshare an edge. Then the weak dual of an outerplanar graph is a forest. When G is2-connected and outerplanar, G∗

w is a tree, and thus when G has at least three faces,it contains an “ear;” an ear is a finite face incident along an edge with just one otherfinite face of the graph.

3. BIPARTITE OUTERPLANAR GRAPHS

First we note that a 1-connected, outerplanar bipartite graph with |L(v)| ≥min{deg(v), 4} need not have an L-list-coloring, and the same holds for outerplanarbipartite graphs with a separating vertex and minimum degree 2. Clearly vertices ofdegree 1 with a 1-list can be placed adjacent to any vertex of degree 4 or more andso destroy the list-colorability of the latter vertex, for example, as in K1,4. Whenthe minimum degree is 2, then let C4 be the 4-cycle, v1, . . . , v4. If L(v2) = {a, b},L(v3) = {b, c}, and with c �= a L(v4) = {a, c}, then C4 cannot be list-colored if v1

is colored with a. Take four C4’s and identify one vertex of each to become a vertexv∗ of degree 8. If L(v∗) = {a, b, c, d}, then appropriately modifying the 2-lists onthe four C4’s creates a graph in which v∗ cannot be colored from its list.

For both Theorems 1 and 3 we shall prove a stronger result, which is needed forthe inductive argument. Since every outerplanar graph contains an ear, notice thatas in Figure 1, a graph G that contains an ear that is a quadrilateral with 2-lists asshown can be list-colored if and only if the graph G′, G minus that quadrilateral,can be list-colored with the coloring (a, b) forbidden on the new external edge. Wedenote this forbidden coloring, an ordered pair, by f (v1, v4) = (a, b), and say thata (respectively, b) is a color forbidden by f (v1, v4) at v1 (resp., v4). Note that sucha forbidden coloring arises in the smaller graph G′ only when for v = v1 and v4,|L(v)| ≥ min{degG′(v) + 1, 4}. This technique of deleting two vertices of a 4-cycleand substituting a forbidden coloring will extend to the case when the ear is a largereven cycle.

We note that the following theorem is more easily proved when G is a 2-connected, outerplanar near-quadrangulation, but we include the proof of the



FIGURE 1. Deletion of a quadrilateral and the resulting forbidden coloring.

general bipartite case here. The desired list-coloring result (of the second sentence)follows by assigning no forbidden coloring.

Theorem 1. Let G be a 2-connected, outerplanar bipartite graph. If |L(v)| ≥min{deg(v), 4} for each vertex v, then G is L-list-colorable. In addition if each ex-ternal edge e = {v1, v2} with |L(vi)| ≥ min{degG(vi) + 1, 4}, i = 1, 2, has at mostone forbidden coloring f (e) = (c1, c2), ci in L(vi) for i = 1, 2, then G is L-list-colorable with the vertices on no external edge e receiving the forbidden coloringf (e) provided that whenever two consecutive external edges {x1, x2} and {x2, x3}both have a forbidden coloring, then degG(x2) ≥ 4.

Proof. We call a vertex v well listed if when it is incident with k = 0, 1, or 2external edges with a forbidden coloring, |L(v)| ≥ min{deg(v) + k, 4}.

When G is an even cycle with each vertex having a 2-list, the graph is list-colorable. Here is a procedure to list-color (in fact, any cycle) G also when someedges have a forbidden coloring, provided that each vertex is well listed. Label thevertices of the graph clockwise, v0, v1, v2, . . . , vn−1, and without loss of generalitysuppose {v0, v1} has a forbidden coloring. Then |L(v1)| ≥ 3 and we select a colorfor v1 that is not forbidden at v1 either on {v0, v1} or on {v1, v2} (if f (v1, v2) isdefined). At v2 there are at least one, two, or three colors available, different thanthe color of v1, if |L(v2)| = 2, 3, or ≥ 4, respectively. In the latter two cases, select acolor for v2 that is not forbidden at v2 on {v2, v3} (if there is such a forbidden); in thefirst case there is no such forbidden coloring since v2 has a 2-list. Continue aroundthe cycle in this manner. When the final vertex v0 is reached, the forbidden coloringson {vn−1, v0} (if defined) and on {v0, v1} do not occur, and since |L(v0)| ≥ 3 there isa color available other than those on vn−1 and v1. Thus the base case of the theoremholds, when the graph has f = 2 faces.

Assume the result is true for every hypothesized graph with f ≥ 2 faces. LetG be a 2-connected, outerplanar bipartite graph with f + 1 faces with forbiddencolorings on some (or none) of the external edges that join two well listed vertices. Gcontains an ear, which is a 2i-sided face, i ≥ 2, (v1, v2, . . . , v2i), where deg(vj) =2 for j = 2, . . . , 2i − 1, and deg(v1), deg(v2i) ≥ 3. Remove the path of verticesP = v2, v3, . . . , v2i−1 (of at least two vertices) to form G′, and note that in all



cases v1 and v2i are well listed vertices in G′ even if f (v1, v2i) becomes assigned,as described below. This is so since for j = 1, 2i, min{degG′(vj) + 1 + kj, 4} =min{degG(vj) + kj, 4} ≤ |L(vj)|, where kj is 0 or 1 according as the external edge{vj, v

′j} (with labels as in Fig. 1) does not or does have a forbidden coloring.

In all cases we remove the vertices of P to form G′, list-color G′ by inductionsince its vertices are well listed, and attempt to extend the coloring, moving forwardsalong the vertices of P. We shall see that there is never trouble in coloring the first2i − 3 vertices of P, but there may be difficulty at v2i−1 which will instruct us onwhat forbidden coloring to assign as f (v1, v2i). Symmetrically we could color thevertices in reverse order along P, and we shall see that the same forbidden coloringis needed as in the forwards case.

For j = 2, . . . , 2i − 2, we will see there is always a color available for vj; its colormust differ from that of vj−1. If f (vj−1, vj) is defined and equals, say, (cj−1, cj),then |L(vj)| ≥ 3, and if vj−1 has been colored with cj−1, then an additional color(cj) is possibly forbidden to vj, but there is another available color in L(vj). Inaddition there is more than one choice for the color of vj when (i) |L(vj)| ≥ 2and vj−1 is not colored with a color of L(vj), (ii) |L(vj)| ≥ 3 and f (vj−1, vj) isnot defined, or when f (vj−1, vj) = (cj−1, cj) and vj−1 is not colored with cj−1 orcj /∈ L(vj), or (iii) when |L(vj)| ≥ 4. Thus if there is only a single available colorat vj and it has a 3-list, then vj is preceded by an edge with a forbidden coloringand followed by one with no forbidden, and if there is only a single available colorat vj and it has a 2-list, then that list overlaps with vj−1’s list. In summary there isa coloring, but no choice in the coloring, of P (up through v2i−2), precisely wheneither (1) v1 is colored with, say, c2 and L(vj) = {cj, cj+1}, j = 2, . . . , 2i − 2,or (2) v1 is colored with c1, f (v1, v2) = (c1, c2), L(v2) = {c1, c2, c3}, and L(vj) ={cj, cj+1}, j = 3, . . . , 2i − 2. In each case what happens when we attempt to extendthe coloring of P to v2i−1?

When L(v2i−1) = {a, b} with, say, color a appearing on v2i−2 and b on v2i, thenthere is no color for v2i−1 precisely when there is no choice along the path P; if thereis choice along P, backtracking to the closest vertex to v2i−1 at which there was achoice, making another choice there and extending, avoiding the previous coloring,will lead to a coloring that allows color a on v2i−1. Suppose L(v2i−1) = {a, b, c}with, say, color a appearing on v2i−2, f (v2i−1, v2i) = (b, c), and c appearing on v2i.As before there is then no color for v2i−1 when there is no choice along the path P,but v2i−1 can be colored with color a when a different choice can be made alongP. In all other cases either there is a choice for v2i−1 or there is choice along Pthat allows for a color for v2i−1. (For example it might seem that when v2i−1 has a3-list and f (v2i−2, v2i−1) is defined or when v2i−1 has a 4-list and is incident withtwo edges having forbiddens, that there might be no color for v2i−1, but by theargument of the preceding paragraph there must be a choice of colors in P leadingto a different color at v2i−2 in all these cases.)

The resulting four cases and the needed forbidden colorings on {v1, v2i} areas follows: Suppose there are no forbidden colorings other than those explicitlymentioned in these cases and for j = 2, . . . , 2i − 2, L(vj) = {cj, cj+1}. Consider



when

(1) v1 is colored with c2, or(2) v1 is colored with c1, f (v1, v2) = (c1, c2), and L(v2) = {c1, c2, c3},

and

(i) L(v2i−1) = {c2i−1, c2i} and v2i is colored with c2i, or(ii) L(v2i−1) = {c2i−1, c2i, c2i+1}, f (v2i−1, v2i) = (c2i, c2i+1), and v2i is colored

with c2i+1.

Then we define in case (1i) f (v1, v2i) = (c2, c2i), in case (1ii) f (v1, v2i) =(c2, c2i+1), in case (2i) f (v1, v2i) = (c1, c2i), and in case (2ii) f (v1, v2i) =(c1, c2i+1).

Now we claim that the list-coloring of G′ extends to G as needed. When there ischoice along P, it is possible to color v2i−1 as well. When there is no choice alongP, we must be in case (1) or (2), and then there is no color for v2i−1 in cases (i)or (ii). In these cases we must color vj with cj+1 for j = 2, . . . , 2i − 2. Due to theforbidden colorings v2i is not colored with c2i in cases (1i) and (2i) (respectively,is not colored with c2i+1 in cases (1ii) and (2ii)) so that v2i−1 can be colored withc2i (resp., with c2i+1).

Notice that the conditions requiring forbidden colorings are symmetric withrespect to the beginning and end of P so that no additional conditions are neededwhen we choose to color P “backwards” in decreasing order of subscripts. Weconclude that G is always list-colorable with forbidden colorings on the externaledges observed when present. �

We note that all conditions of the hypotheses of Theorem 1 are needed; theneed for 2-connectivity was shown above. If G is 2-connected, K4-minor-free,and bipartite with |L(v)| ≥ min{deg(v), 4} for each vertex v, the graph may notbe L-list colorable. Consider twelve 4-cycles C4 (v1, v2, v3,i, v4,i), i = 1, . . . , 12,that share the common edge {v1, v2}. For i = 1, 2 let L(vi) = {a, b, c, d}, and fori = 1, . . . , 12 let L(v3,i) = {x, z}, and L(v4,i) = {y, z}, where (x, y), x �= y, variesover all 12 ordered pairs taken from {a, b, c, d}.

It is not hard to find examples of 2-connected, outerplanar near-triangulationswith |L(v)| = min{deg(v), 3} that are not L-list colorable, but such a bipartite exam-ple seems more difficult. The next example shows that the lower bound of 4 cannotbe replaced by 3; that is, in Figure 2iii is an outline of a 2-connected, outerplanarbipartite graph in which degree-2 vertices have 2-lists and all the rest have 3-lists,but the graph is not list-colorable with these lists. First notice that vertex v∗ in thegraph of Figure 2i cannot be colored with color a. Then in the graph in Figure 2iiv∗ cannot be colored with a or b, with the same lists as in Figure 2i in the top halfof the graph.



FIGURE 2. (i) Vertex v∗ cannot be colored with a; (ii) v∗ cannot be colored with aor b; (iii) An outline of a non-list-colorable graph.

Thus v∗ must be colored by c and the corresponding vertex w∗ in the flip of thegraph across a vertical line must also be colored with c, giving the impossibility oflist-coloring the full graph of Figure 2iii.

The even more important example of Figure 3 shows that if G is 2-connected,outerplanar, and has |L(v)| ≥ min{deg(v), 4} for each vertex v, it may not be L-list-colorable [11].



FIGURE 3. A non-list-colorable graph.

To see that this graph is not list-colorable, color a degree-6 vertex with color aand one neighbor on the outer face with b. Then try to extend moving around thecycle in the direction of the b-colored vertex. A similar demonstration works whenthe initial degree-6 vertex is colored with b, c, or d.

This example fuels much of our work in Section 4, and a similar 3-connectedexample is displayed in Section 5. Kostochka’s example led to the following con-jecture, which has been proven in [8].

Conjecture [10]. If G is a 2-connected, outerplanar near-triangulation with atmost three internal triangles and if |L(v)| ≥ min{deg(v), 4} for each vertex v, thenG is L-list-colorable.

4. GENERAL OUTERPLANAR GRAPHS

We turn now to the general case and focus on 2-connected, outerplanar near-triangulations to show that when |L(v)| ≥ min{deg(v), 5} for all v, the graph canbe L-list-colored unless it is K3 with all 2-lists identical. Note that 5 cannot bereplaced by 4 due to Kostochka’s example of Figure 3. Also 1-connectivity is notsufficient, as seen by considering K1,5, and 1-connectivity with minimum degree2 is not sufficient due to disjoint triangles sharing a vertex; these are Gallai trees.And the list-coloring result does not hold for 2-connected K4-minor-free graphs,for consider 10 triangles {v1, v2, v3,i}, i = 1, . . . , 10, that share the common edge{v1, v2}. For i = 1, 2 let L(vi) = {a, b, c, d, e}, and let the lists on the v3,i compriseall 2-subsets of {a, b, c, d, e}.

The proof-technique uses more complex forbidden colorings on external edgesthan in Theorem 1, and as before an extended result with these is derived. First



FIGURE 4. Deletion of a triangle ear and the resulting forbidden colorings.

note that a graph with a triangle ear as shown in Figure 4 can be list-colored ifand only if the colorings (a, b) and (b, a) are forbidden on the new external edgee′ = (v1, v3) when the triangle ear is deleted. We denote these two colorings byf (e′) = {a, b}, and note that when such a forbidden coloring is defined on G′, thenwe must have for i = 1, 3|L(vi)| ≥ min{degG′(vi) + 1, 5}. We say that colors a andb are forbiddens at vi of f (e′), i = 1, 3.

Drawing from Kostochka’s example, we note that a graph with two successivetriangle ears as shown in Figure 5 can be list-colored if and only if the colorings(a, c) and (b, c) are forbidden on the new external edge (v1, v4). We denote thesetwo colorings on e′ = (v1, v4) by f (e′) = (a, b; c) where order matters, and notethat when such a forbidden coloring is defined on G′, then we must have |L(v1)| ≥min{degG′(v1) + 2, 5} and |L(v4)| ≥ min{degG′(v4) + 1, 5} . We say that colors aand b (respectively, c) are forbiddens at v1 (resp., v4) of f (e′).

For the final set of forbidden colorings, we note that a graph with three successivetriangle ears as shown in Figure 6 can be list-colored if and only if the colorings (a,c), (b, c), (a, d), and (b, d) are forbidden on the new external edge e′ = (v1, v5). Wedenote these colorings on e′ by f (e′) = (a, b; c, d), where order matters, and notethat for i = 1, 5, we must have |L(vi)| ≥ min{degG′(vi) + 2, 5}. We say that colorsa and b (respectively, c and d) are forbiddens at v1 (resp., v5) of f (e′).

Notice that in each case and at each vertex at most four colors are forbidden bythe two incident external edges.

Finally notice that if the external edges (v1, v2) and (v2, v3) are consecutive andboth have forbidden colorings in G, then we must also have |L(v2)| ≥ 5 unless



FIGURE 5. Deletion of two triangle ears and the resulting forbidden colorings.

f (v1, v2) = {a, b} or (a, b; c) and f (v2, v3) = {d, e} or (d; e, f) for some choices ofcolors, and in these cases we must have |L(v2)| ≥ min{degG(v2) + 2, 5}.

Suppose G is a 2-connected, outerplanar near-triangulation with some externaledges having two, three, or four forbidden colorings as described above. In thiscontext we say that a vertex is well listed provided it meets the above list-size

FIGURE 6. Deletion of three triangle ears and the resulting forbidden colorings.



criteria. Precisely the vertex v is well listed provided (i) when v is incident withno external edge with forbidden colorings |L(v)| ≥ min{degG(v), 5}, (ii) when v isincident with exactly one external edge e = (u, v) with f (e) = {a, b} or (a, b; c),then |L(v)| ≥ min{degG(v) + 1, 5}, (iii) when v is incident with two external edgese = (u, v) and e′ = (v, w) with f (e) as in case (ii) and f (e′) = {d, e} or (d; e, f), orwhen v is incident with exactly one external edge e = (u, v) so that f (e) forbidstwo colors on v (precisely when f (e) = (a; b, c) or (a, b; c, d) for some colors a, b,c, and d), then |L(v)| ≥ min{degG(v) + 2, 5}, and iv) otherwise |L(v)| ≥ 5.

Using these sorts of reductions and forbidden colorings we check the base casewith three vertices. Notice that Kostochka’s example of Figure 3 shows that K3 withthe four forbidden colorings (a, b; c, d) on each edge would not be list-colorable,but in that case of the next lemma, we require the lists to be each of size at leastfive, producing a graph that can be list-colored avoiding all the forbiddens.

Lemma 2. Suppose G is a triangle {v1, v2, v3}, each edge may have two, threeor four forbidden colorings, and the lists on vertices create all well listed vertices.Then G can be list-colored provided the lists are not three identical 2-lists.

Proof. When no edge has a forbidden coloring, it is easy to check that Gcan be list-colored except in the exceptional case. Let the edges of G be labeledei = {vi, vi+1}, i = 1, 2, 3, with subscripts read modulo 3.

Suppose G has precisely one edge, say e1, with f (e1) defined. Then |L(vi)| ≥ 3,i = 1, 2, and regardless of the value of f (e1) we can choose a color for v1 that isnot forbidden at v1. There is a color for v3 and then one for v2 since f (e1) hasbeen avoided at v1. Next suppose precisely f (e1) and f (e2) are defined so that|L(vi)| ≥ 3, i = 1, 3, and |L(v2)| ≥ 4. If possible, select a color for v2 that is notforbidden at v2 by either edge; then the coloring extends as in the previous case.If not possible, v2 has a 4-list equaling the set of four colors forbidden at v2; picka color for v2 that is not a forbidden of f (e1) at v2, though it will be forbiddenby f (e2). Since v2 has a 4-list, not a 5-list, and f (e2) forbids two colors there,f (e2) = {a, b} for some two colors. Then at v3 with a 3-list, there is a color that isneither a nor b, and this coloring extends to v1.

Finally suppose all three edges have forbiddens defined and thus all lists have atleast four colors. If one vertex, say v2, has a color in its list that is not forbiddenon either incident edge, we use that color. This extends to v3, avoiding the colorof v2 and at most two forbidden colors of f (e3). This coloring extends to v1 asabove. Otherwise each vertex has four forbidden colors (the maximum possible)and each list is identical to the set of four forbidden colors. Then, as in the previousparagraph, we pick a color for v2 that is not forbidden by f (e1) but is forbidden byf (e2). Then at v3 we pick a color that is not that of v2 nor is forbidden by f (e2).Now since v1 and v3 have a 4-list, not a 5-list, f (e3) must be of the form {a, b}, andv3 is colored with a forbidden color of f (e3), say, color a. Then there is a color inL(v1) that is not a, b, or the color of v2. �Journal of Graph Theory DOI 10.1002/jgt


We know from [4,5,7,17,19] that a graph G with |L(v)| = deg(v) for every ver-tex v is list-colorable if and only if G is not a Gallai tree. Since we consideronly 2-connected, outerplanar graphs, the only graphs with maximum degree atmost 5 that might be a counterexample are the odd cycles. Since we considernear-triangulations, it is only K3 with all 2-lists identical that is an exception toTheorem 3.

Theorem 3. Let G be a 2-connected, outerplanar near-triangulation with atleast three vertices that is not K3 with identical 2-lists at each vertex. If |L(v)| ≥min{deg(v), 5} for every vertex v, then G is list-colorable. In addition if some ex-ternal edges have forbidden colorings and the incident vertices are all well listed,then G is L-list-colorable with vertices on no external edge e receiving a forbiddencoloring of f (e).

Proof. The proof is by induction on n, the number of vertices of G. Lemma 2establishes the base case when n = 3. Let G meet the hypothesis with n > 3. SinceG is a near-triangulation, we choose a vertex v2 of degree 2; we assume that thevertices of G are labeled clockwise around the external face v1, v2, . . . , vn, and letei = {vi, vi+1}, for i = 1, . . . , n with subscripts read modulo n. In each case wedelete e1, v2, and e2 to form G′ and let e′ = (v1, v3) be the new external edge, asshown in Figure 4 (though f (e′) may be assigned a different value, as describedbelow). Notice that since degG(v1) and degG(v3) are at least three, their lists aresimilarly large. In particular G′ is not the triangle with vertices having identical2-lists.

If both f (e1) and f (e2) are undefined and L(v2) = {a, b}, then set f (e′) = {a, b},and otherwise f (e′) is not defined. Notice that in G′ both v1 and v3 are well listedwhenf (e′) is defined (or not). We have |L(v1)| ≥ min{degG(v1) + k, 5}, with k = 0,1, or 2 depending on the value of f (en), and thus |L(v1)| ≥ min{degG′(v1) + 1 +k, 5} = min{degG(v1) + k, 5}. The same holds for v3, and thus the list-sizes allowf (e′) to be set to {a, b} in G′. Then G′ is list-colorable avoiding forbidden coloringson external edges by induction, and this coloring extends to v2 and so to G.

When only f (e1) is defined, we have four possible values to consider. Inall four cases due to the assignment of f (e′), v3 will remain well listed since|L(v3)| ≥ min{degG(v3) + k, 5} = min{degG′(v3) + 1 + k, 5}, where k = 0, 1, or2 depending on f (e3). When f (e1) = {a, b}, then |L(v2)| ≥ 3, and |L(v1)| ≥min{degG(v1) + 1 + k, 5} = min{degG′(v1) + 2 + k, 5}, where k = 0, 1, or 2 de-pends on f (en). In this case when L(v2) = {a, b, c}, we set f (e′) = (a, b; c) andotherwise it is undefined. Similarly when f (e1) = (c; a, b), then |L(v2)| ≥ 4 and|L(v1)| is as in the previous case. In this case when L(v2) = {a, b, c, x}, we setf (e′) = {c, x} and is otherwise undefined. As before G′ with f (e′) is well listed.

When f (e1) = (a, b; c) , then |L(v2)| ≥ 3, and |L(v1)| ≥ min{degG(v1) +2, 5} = min{degG′(v1) + 3, 5} = 5. Then when L(v2) = {a, c, x} (respectively,{b, c, x}), we set f (e′) = {a, x} (resp., {b, x}) and otherwise is undefined. Whenf (e1) = (a, b; c, d), |L(v2)| ≥ 4 and |L(v1)| is as in the previous case. In this casewhen L(v2) = {a, c, d, x} (respectively, {b, c, d, y}) we set f (e′) = {a, x} (resp.,



{b, y}) and otherwise is undefined. Thus in all cases G is transformed to G′, whichis well listed, is list-colored avoiding forbiddens on external edges by induction,and the coloring extends to G. (A close examination of these cases shows that some-times just one coloring, like (a, x), needs to be excluded, but we use the strongerrequirement that {a, x} be excluded.)

Finally suppose that both f (e1) and f (e2) are defined. There are 16 possi-ble forbidden colorings, which we deal with in batches. In all cases we havethat |L(v2)| ≥ 4 so that when v1 and v3 are colored by induction, there are atleast two colors for v2 unless those colors are involved in forbiddens. We havealso for i = 1, 3|L(vi)| ≥ min{degG(vi) + 1 + ki, 5} = min{degG′(vi) + 2 + ki, 5}where k1 (respectively, k3) depends upon the value of f (en) (resp., f (e3)). SupposeL(v2) contains {w, x, y, z} and that two of these colors, say y and z, are not forbid-dens at v2. Then in G′ we assign f (e′) = {y, z}; since G′ is well listed it can belist-colored by induction and the coloring extends to G.

Otherwise at most one of {w, x, y, z} is not a forbidden color at v2, and when one,suppose it is z. We have that |L(v2)| = 4 or 5 (since with 6 or more, the previouscase holds.) First assume that L(v2) = {w, x, y, z}. Due to the assumptions of thiscase, without loss of generality f (e1) = {w, x} and the other forbiddens at v2 off (e2) are (i) y, (ii) x and y, or (iii) y and z. There is a variety of cases to consider,but only a few require that f (e′) be defined. Specifically, in case (i) only whenf (e2) = (y; c, d) with z = c (or d), then f (e′) must be defined to be (w, x; z). Incase (ii) v2 can always be colored, and in case (iii) only when f (e2) = {y, z}, thenf (e′) must be defined to be (w, x; y, z). As above v1 and v3 are well listed in G′

so that G′ can be list-colored avoiding forbiddens by induction and the coloringextends to G.

Next assume that L(v2) = {u, w, x, y, z}, u and w are forbiddens at v2 of f (e1),x and y are forbiddens at v2 of f (e2), and z is not a forbidden color. Since L(v2) hasfive elements, we may assume without loss of generality that (i) f (e1) = (a; u, w) or(ii) f (e1) = (a, b; u, w) for some colors a and b. Then we must have f (e2) = {x, y},and the only cases that require an assignment to f (e′) are (i) when z = a or (ii)z = a or b, and in these cases we let f (e′) = (z; x, y). In each of these cases thesame arguments show that G′ is well listed, can be colored by induction and thecoloring extends to v2 and so to G. �

When a 2-connected, outerplanar near-triangulation has no forbidden coloringsassigned, then the basic list-coloring result follows when |L(v)| ≥ min{deg(v), 5}for each vertex v.

5. SOME CONCLUDING QUESTIONS

Richter [15] has asked if a similar list-coloring result holds for all planar graphs.If G is a 3-connected planar graph and if |L(v)| ≥ min{deg(v), 6} for every vertexv, is G L-list-colorable? We have seen an example that shows 2-connectivity is notsufficient for this result even with the 6 replaced by k ≥ 6: consider

(k

2

)triangles



FIGURE 7. An outline of a non-list-colorable graph.

that share one edge with each 2-subset of a k-set appearing on one degree-2 vertexand the k-set on both vertices of degree

(k

2

) + 1. Also being K5-minor-free, insteadof planar, is not sufficient due to the example of a triangle with each vertex havingthe same k-list and to which

(k

3

)degree-3 vertices are adjacent, each with a distinct

triple taken from the k-list.For an example of a 3-connected graph which with appropriate lists L, |L(v)| ≥

min{deg(v), 4}, is not L-list-colorable, take a non-4-choosable graph of [18] or[13] and decrease the list on each vertex of degree 2 or 3 to one of size 2 or 3respectively; the resulting graph remains non-list colorable. For a simpler example,Figure 7 shows (the start of) a 3-connected graph with |L(v)| ≥ min{deg(v), 4} thatis not L-list-colorable; give each vertex the list {a, b, c, d}. Then add one vertex ofdegree 3 to each of the six faces. Give the vertices in the three faces incident with 1the list {a, b, c} and those in the three faces incident with 5 the list {a, b, d}. Thenonly colors {a, b} are available for the three vertices 2, 3, and 4, for if one, say 2,were colored c (or d), then the triangle {3, 4, 5} (or {1, 3, 4}) must be colored with{a, b, d} (or {a, b, c}), leaving no color for the interior degree-3 vertex.

The answer to Richter’s question is yes with 5-connectivity since such graphshave minimum degree 5 and all planar graphs are 5-choosable [16]. It is unknownwhether this question is true for 4-connected planar graphs. The answer is yes alsoif G is a 3-connected planar graph with |L(v)| ≥ min{deg(v), 6} and with every pairof vertices of degree less than 6 having mutual distance at least three, since if eachsuch small-degree vertex is assigned a color from its list and that color is deletedfrom all neighboring vertices (of degree at least 6), then the remaining graph is5-list-colorable by [16]. Perhaps an easier question to settle would be the analogue



of outerplanar graph, the case of planar 3-trees. These are planar graphs formedfrom an initial K3 with sequential additions of degree-3 vertices joined to the threevertices of a K3 of the previous graph; the enhanced graph from Figure 7 is a 3-tree.

ACKNOWLEDGMENTS

I thank the following people for very helpful conversations on this work: M.O.Albertson, A. Kostochka, E.H. Moore, R. Ramamurthi, B. Richter, M. Steibitz, C.Thomassen, and M. Voigt.

REFERENCES

[1] M. O. Albertson, You can’t paint yourself into a corner, J Combin Theory SerB 78 (1998), 189–194.

[2] M. O. Albertson and J. P. Hutchinson, Graph color extension: when Hadwiger’sConjecture and embeddings help, Electron J Combin 9 (2002), R37 (10 pages).

[3] M. O. Albertson and E. H. Moore, Extending graph colorings, J CombinTheory Ser B 77 (1999), 83–95.

[4] O. V. Borodin, Criterion of chromaticity of a degree prescription, In: Abstractsof IV All-Union Conf. on Theoretical Cybernetics (Novosibirsk) (1977),pp. 127–128 (in Russian).

[5] O. V. Borodin, Problems of colouring and of covering the vertex set of a graphby induced subgraphs. Ph.D. Thesis, Novosibirsk State University, Novosi-birsk, 1979 (in Russian).

[6] R. Diestel, Graph Theory, Springer-Verlag, New York, 1997.[7] P. Erdos, A. L. Rubin, and H. Taylor, Choosability in graphs, In: Proc. West

Coast Conf. on Comb., Graph Theory, and Computing, Arcata, CA, 1979,Congressus Numerantium XXVI (1979), pp. 125–157.

[8] S. Fried and E. H. Moore, On list-coloring 2-connected, outerplanar near-triangulations (manuscript, 2007).

[9] J. P. Hutchinson and E. H. Moore, Distance constraints in graph color exten-sions, J Combin Theory Ser B 97 (2007), 501–517.

[10] J. P. Hutchinson and R. Ramamurthi, personal communication.[11] A. V. Kostochka, personal communication.[12] A. V. Kostochka, M. Stiebitz, and B. Wirth, The colour theorems of Brooks

and Gallai extended, Discrete Math 162 (1996), 299–303.[13] M. Mirzakhani, A small non-4-choosable planar graph, Bull Inst Combin Appl

17 (1996), 15–18.[14] A. Pruchnewski and M. Voigt, Precoloring extension forK4-minor-free graphs,

J Graph Theory (submitted).[15] A. B. Richter, personal communication.



[16] C. Thomassen, Every planar graph is 5-choosable, J Combin Theory Ser B 62(1994), 180–181.

[17] C. Thomassen, Color-critical graphs on a fixed surface, J Combin Theory SerB 70 (1997), 67–100.

[18] M. Voigt, List colourings of planar graphs, Discrete Math 120 (1993), 215–219.

[19] V. G. Vizing, Colouring the vertices of a graph with prescribed colours, MetodyDiskret Analiz 29 (1976), 3–10 (in Russian).

[20] D. B. West, Introduction to Graph Theory, Second Edition, Prentice-Hall,Upper Saddle River, NJ, 2001.


on list-coloring outerplanar graphs

Documents