Digital Object Identifier (DOI) 10.1007/s00373-005-0645-9Graphs and Combinatorics (2006) 22:173–183

Graphs andCombinatorics© Springer-Verlag 2006

Sum List Coloring Graphs

Adam Berliner, Ulrike Bostelmann, Richard A. Brualdi and Louis DeaettDepartment of Mathematics, University of Wisconsin, Madison, WI 53706 USA.e-mail: berliner@math.wisc.edu,bostelma@math.wisc.edu,brualdi@math.wisc.edu,deaett@math.wisc.edu

Abstract. Let G = (V , E) be a graph with n vertices and e edges. The sum choice number ofG is the smallest integer p such that there exist list sizes (f (v) : v ∈ V ) whose sum is p forwhich G has a proper coloring no matter which color lists of size f (v) are assigned to thevertices v. The sum choice number is bounded above by n + e. If the sum choice number ofG equals n + e, then G is sum choice greedy. Complete graphs Kn are sum choice greedy asare trees. Based on a simple, but powerful, lemma we show that a graph each of whose blocksis sum choice greedy is also sum choice greedy. We also determine the sum choice numberof K2,n, and we show that every tree on n vertices can be obtained from Kn by consecutivelydeleting single edges where all intermediate graphs are sc-greedy.

Key words. Graph, Choosable function, List coloring, Sum choice number, Sum choicegreedy graph

1. Introduction

Let G = (V , E) be a graph with vertex set V and edge set E. List coloring is ageneralization of graph coloring whereby the color of each vertex is taken from apre-assigned list of possible colors for the vertex. Let C = (Cv : v ∈ V ) be a familywhere Cv is a nonempty set of elements called colors assigned to vertex v, (v ∈ V ).A C-list coloring of G is a function f : V → ∪v∈V Cv such that f (v) ∈ Cv for eachv ∈ V and f (u) �= f (w) for each pair of distinct vertices u and w joined by an edgein E. Thus list coloring is a generalization of classical coloring where it is assumedthat each vertex has the same set of available colors. At first glance, a list coloringseems easier to construct, since there are, in general, more colors to choose from.However, as is well known, in the case of the complete bipartite graph K3,3 (sochromatic number equals 2) with vertex partition V1 = {x, y, z} and V2 = {p, q, r},and color lists Cx = Cp = {R, B}, Cy = Cq = {R, Y } and Cz = Cr = {B, Y } ofsize 2, there is no list coloring. In this case, the structure of the color lists forces twodifferent colors to be used in coloring the vertices in V1 and this prevents one of thevertices in V2 being colored. Thus, if a graph G has chromatic number k, one cannotbe assured of a list coloring of G even when all the color lists Cv have size k.

Let χ(G) denote the chromatic number of G. The list chromatic number χl(G)

is the smallest number t such that a C-list coloring exists whenever the family C =

174 A. Berliner et al.

(Cv : v ∈ V ) satisfies |Cv| ≥ t for all v ∈ V . We have χl(G) ≥ χ(G) and, as theabove example shows, strict inequality is possible. For a recent survey paper on listcoloring and for many references, one can consult [5].

Let f : V → Z+. A family C = (Cv : v ∈ V ) is called an f -family provided that|Cv| = f (v) for each v in V . The graph G is called f -choosable provided G has aC-list coloring for all f -families C. If G is f -choosable, then f is a choice functionfor G. Thus K3,3 is not f -choosable for the constant function f = 2 on its vertices.Isaak [3] has defined the sum choice number χsc(G) of G by

χsc(G) = minf

∑

v∈V

f (v) (1)

where the minimum is taken over all functions f such that G is f -choosable. Achoice function f for G satisfying (1) is a minimum choice function for G. By divid-ing by the number of vertices, we get the average choice number of G. Notice that iff is a minimum choice function, then for each vertex v, f (v) is at most one morethan the degree of v.

We briefly summarize results of Isaak in [3] and [4]. For the line graph L(K2,n)

of the complete bipartite graph K2,n (perhaps best viewed as a 2 by n array ofsquares—the vertices—with two squares joined by an edge if and only if they are inthe same row or the same column), we have χ(L(K2,n)) = n2 +�5n/3�. A large partof the proof is in showing that L(K2,3) is f -choosable for the function f depictedas:

2 2 33 2 2 .

In [3] it is remarked that the sum choice number of L(K3,3)—a 3 by 3 array—hasbeen verified to be either 24 or 25. (By a case by case analysis, too tedious to givehere, we have shown that the sum choice number is in fact 25.)

Let n be the number of vertices and let e be the number of edges of G. Letdv denote the degree of a vertex v. Let the vertices of G be listed in some orderv1, v2, . . . , vn, and let d ′

viequal the number of vertices among v1, . . . , vi−1 that are

joined to vi by an edge. Define a function f by

f (vi) = d ′vi

+ 1, (i = 1, 2, . . . , n).

A greedy algorithm—assuming, without loss of generality, that the color lists aretaken from a linearly ordered set and always choosing the smallest available color—shows that G is f -choosable. Since

∑ni=1 f (vi) = n + e, we have

χsc(G) ≤ n + e.

A graph G for which χsc(G) = n+e will be called a sc-greedy graph. In [4] it is shownby a simple argument that graphs whose blocks are complete graphs are sc-greedygraphs, thus giving a formula for their sum choice numbers. In particular, completegraphs, trees, and line graphs of trees are sc-greedy graphs. It is also easy to verifythat cycles are sc-greedy.

Sum List Coloring Graphs 175

We now briefly summarize the main results in this paper. Based on a simple butpowerful lemma, we give a simple proof that a graph whose blocks are sc-greedygraphs is an sc-greedy graph. A special case is Isaak’s theorem that a graph whoseblocks are complete graphs is a sc-greedy graph. We show that a graph obtainedfrom a complete graph Kn by adjoining a new vertex joined to some of the verticesof Kn is a sc-greedy graph. As a consequence, we show that given any tree Tn onn vertices, there is a sequence of graphs starting with Kn and ending with Tn, suchthat each graph other than the first is obtained from the preceding one by removingan edge and each graph in the sequence is a sc-greedy graph. We also determine thesum choice number of the complete bipartite graph K2,n.

2. Sum Choice Number

We begin with the following elementary, yet powerful, lemma.

Lemma 1. Let G = (V , E) be a graph and let f : V → Z+ be a function such that G

is f -choosable. Let r be the nonnegative integer with

∑

v∈V

f (v) = χsc(G) + r.

Let u be a vertex of G. Then for each set S of r + 1 colors, there exists an f -familyC = (Cv : v ∈ V ) such that in every C-list coloring of G, vertex u receives a color in S.

Proof. If f (u) ≤ r + 1, then any f -family C = (Cv : v ∈ V ) with Cu ⊆ S satisfiesthe conclusion of the lemma. Now assume that f (u) ≥ r + 2. Let f ′ : V → Z+be defined by f ′(u) = f (u) − (r + 1) and f ′(v) = f (v) for all v �= u. Since∑

v∈V f ′(v) = χsc(G) − 1, there is an f ′-family C′ = (C′v : v ∈ V ) such that G does

not have a C′ list coloring. Changing names of colors as needed, we may assumethat S ∩ C′

u = ∅. Now let C = (Cv : v ∈ V ) be defined by Cu = C′u ∪ S and Cv = C′

v

for v �= u. Then C is an f -family and hence G has a C-list coloring. Clearly, in everyC-list coloring of G, vertex u receives a color from S. �

Informally, Lemma 1 says that we can force u to be one of r + 1 colors of ourchoice. The following special case of Lemma 1 is very useful in determining the sumchoice numbers of certain graphs.

Corollary 1. Let G = (V , E) be a graph and let f be a minimum choice functionfor G. Let c be a color and let u be a vertex of G. Then there exists an f -familyC = (Cv : v ∈ V ) such that in every C-list coloring of G, vertex u receives the color c.

If G and G′ are graphs with disjoint vertex sets, then their union G∪G′ satisfiesχsc(G ∪ G′) = χsc(G) + χsc(G

′). We now determine the sum choice number ofG ∪ G′ in case that G and G′ have exactly one vertex in common.

176 A. Berliner et al.

Theorem 1. Let G = (V , E) and G′ = (V ′, E′) be graphs such that V ∩ V ′ = {u}.Then

χsc(G ∪ G′) = χsc(G) + χsc(G′) − 1.

Proof. By constructing a choice function for G ∪ G′, we first show that

χsc(G ∪ G′) ≤ χsc(G) + χsc(G′) − 1. (2)

Let f be a minimum choice function for G, and let f ′ be a minimum choice functionfor G′. Define a function h : V ∪ V ′ → Z+ by

h(x) =

f (x), if u �= x ∈ V ,

f ′(x), if u �= x ∈ V ′,f (x) + f ′(x) − 1, if x = u.

We have∑

x∈V ∪V ′ h(x) = χsc(G) + χsc(G′) − 1. Let C = (Cx : x ∈ V ∪ V ′) be an

h-family. Let hV be the restriction of h to V , and let CV = (Cx : x ∈ V ). We havehV (u) = f (u) − 1 + f ′(u) and

∑v∈V hV (v) = χsc(G) + f ′(u) − 1. Then G has a

CV -list coloring. Let A be the set of possible colors for u in CV -list colorings of G. Weclaim that |A| ≥ f ′(u). Otherwise, let C∗

V = (C∗x : x ∈ V ), where C∗

u = Cu \ A andC∗

x = Cx for x ∈ V \ {u}. We then have |C∗u| ≥ f (u) and |C∗

x | = f (x) for x ∈ V \ {u}with G not having a C∗

V -list coloring, and this contradicts the assumption that f

is a choice function for G. Thus |A| ≥ f ′(u). Now let C′V ′ = (C′

x : x ∈ V ′) whereC′

u = A and C′x = Cx for x ∈ V ′ \ {u}. Since f ′ is a choice function for G′, G′ has a

C′V ′ -list coloring. It follows that G ∪ G′ has a C-list coloring and that h is a choice

function for G ∪ G′. Hence (2) holds.We now show that equality holds in (2). Suppose to the contrary that for some

choice function g : V ∪V ′ → Z+ for G∪G′,∑

x∈V ∪V ′ g(x) = χsc(G)+χsc(G′)−2.

Let gV be the restriction of g to V . Then G is gV -choosable and∑

x∈V g(x) =χsc(G)+m for some nonnegative integer m. By Lemma 1 there is a family C = (Cv :v ∈ V ) of color lists such that in every C-coloring of G, the color of u is taken froma set S of size m + 1.

Let g′ : V ′ → Z+ be defined by

g′(x) ={

m + 1, if x = u,

g(x), otherwise.

Since G ∪ G′ is g-choosable, G′ is g′-choosable. Thus

χsc(G′) ≤

∑

x∈V ′g′(x) = m + 1 +

(∑

x∈V ′g(x)

)− g(u). (3)

We also have∑

x∈V ′g(x) =

∑

x∈V ∪V ′g(x) −

∑

x∈V

g(x) + g(u)

= χsc(G) + χsc(G′) − 2 − (χsc(G) + m) + g(u)

= χsc(G′) − (m + 2) + g(u). (4)

Sum List Coloring Graphs 177

Combining (3) and (4), we get

χsc(G′) ≤ m + 1 +

∑

x∈V ′g(x) − g(u)

= m + 1 + χsc(G′) − (m + 2) + g(u) − g(u)

= χsc(G′) − 1,

a contradiction. Hence equality holds in (2). �

A consequence of Theorem 1 is that the sum choice number of a graph is deter-mined by the sum choice numbers of its blocks (2-connected components).

Corollary 2. A graph each of whose blocks is a sc-greedy graph is also a sc-greedygraph.

Proof. By induction, it is enough to show, in the setting of Theorem 1, that if G andG′ are sc-greedy, then so is G ∪ G′. But this follows since G ∪ G′ has one less vertexthan the number of vertices of G and G′ combined, and the same number of edges.

Since a complete graph is sc-greedy, we obtain the main result of Isaak [4].

Corollary 3. A graph each of whose blocks is a complete graph is sc-greedy.

Since a cycle is also a sc-greedy graph, a graph each of whose blocks is either acomplete graph or a cycle is also sc-greedy. In particular, a tree is sc-greedy.

Example 1. Consider a graph G obtained from a cycle Ck of length k by attachinga graph Gk at each vertex vk of Ck. Then χsc(G) = ∑k

i=1 χsc(Gk) + k. This followsimmediately from k applications of Theorem 1 by starting with Ck and using thefact that χsc(Ck) = 2k. In particular, if each of the Gi is sc-greedy (e.g. completegraphs, trees, cycles), then G is sc-greedy.

Since a tree on n vertices is sc-greedy, its sum choice number equals 2n − 1. Wenow characterize the minimum choice functions of trees with n vertices.

Corollary 4. Let V = {v1, v2, . . . , vn} be a set of n elements, and let a1, a2, . . . , an

be a sequence of n positive integers. Let f : V → Z+ be defined by f (vi) = ai fori = 1, 2, . . . , n. Then f is a minimum choice function for some tree with vertex set V

if and only if

n∑

i=1

ai = 2n − 1. (5)

178 A. Berliner et al.

Proof. The case n = 1 is trivial. If n ≥ 2, then (5) implies that some ak > 1, saya1 > 1. Then a1 − 1, a2, . . . , an are positive integers that sum to 2(n − 1) and thereexists a tree Tn with degree sequence a1 −1, a2, . . . , an. For any such tree Tn we havethe following: Let C = (Cvi

: i = 1, 2, . . . , n) be an f -family. Rooting Tn at vertexv1 and recursively choosing a color of a non-root vertex of degree 1, we obtain aC-list coloring of Tn. Hence Tn is f -choosable. The converse follows from the factthat a tree is sc-greedy. (As pointed out by a referee, the “recursive” idea used in thisproof is contained in one of the first papers on list coloring [1].) �

In general, the determination of the sum choice number of a graph seems tobe a difficult computational problem. With the assumption that all the color listsare restricted to be initial color lists of the form {1, 2, . . . , a}, then determining theminimum sum is NP-complete [2] (but this does not imply that determination of thesum choice number is NP-complete). We now determine the sum choice number ofa family of graphs that are not sc-greedy.

Theorem 2. Let n ≥ 1 be an integer. Then the sum choice number of the completebipartite graph K2,n satisfies χsc(K2,n) = 2n + α where

α = min{k + l : kl > n, with k and l integers}.

Proof. Let the vertex set of K2,n be {u, v, x1, x2, . . . , xn} where u and v are joinedto each of x1, x2, . . . , xn. We first show that if k and l are integers with kl > n,then χsc(K2,n) ≤ 2n + k + l. Define a function g by g(xi) = 2 for i = 1, 2, . . . , n,g(u) = k, and g(v) = l. Let C be a g-family of colors. If Cu and Cv are not disjoint,then, since the color lists Cxi

have size 2, by choosing the same color for u and v wesee that there exists a C-list coloring. Now suppose that Cu ∩ Cv = ∅. Then thereare kl possible pairs for the colors of u and v. Since kl > n, one of these pairs isdifferent from all the Cxi

. Choosing this pair for the colors of u and v leaves a coloravailable for each xi . Hence K2,n is C-list colorable and thus K2,n is g-choosable. Itfollows that χsc(K2,n) ≤ 2n + α.

We now prove that χsc(K2,n) ≥ 2n + α. First we show that we may assume thatf (xi) ≥ 2 for all xi . Let f be a function such that K2,n is f -choosable, and supposethat there is an xi such that f (xi) = 1. Let

a = |{xi : f (xi) = 1}| and b = |{xi : f (xi) = 2}|.

Thus a ≥ 1 and a + b ≤ n. Without loss of generality we assume that f (xn−a+1) =· · · = f (xn) = 1. Consider the subgraph K2,n−a induced on {u, v, x1, . . . , xn−a}.Define a function f ′ by f ′(xi) = f (xi) for i = 1, 2, . . . , n − a, f ′(u) = s − a wheres = f (u), and f ′(v) = t−a where t = f (v). Since K2,n is f -choosable, s ≥ a+1 andt ≥ a + 1. Let C′ be an f ′-family. We construct an f -family C by letting Cxi

= {ci}for i = n−a+1, . . . , n where the ci are a distinct colors not appearing in any of thelists of C′, and adding the colors cn−a+1, . . . , cn to C′

u and C′v. Since K2,n is f -choo-

sable, K2,n has a C-list coloring, and by construction, it contains a C′-list coloringof K2,n−a . Hence K2,n−a is f ′-choosable. Since f ′(xi) > 1 for i = 1, 2, . . . , n−a, it

Sum List Coloring Graphs 179

follows using an argument similar to the one used in the preceding paragraph that

(s − a)(t − a) = f ′(u)f ′(v) > b. (6)

In this situation, we show that there is another function h such that K2,n is (i)h-choosable, (ii) h(xi) ≥ 2 for i = 1, 2, . . . , n, and (iii)

∑v∈V h(v) = ∑

v∈V f (v).We define h by

h(xi) = f (xi) ≥ 2, (i = 1, 2, . . . , n − a),

h(xi) = f (xi) + 1 = 2, (i = n − a + 1, . . . , n),

h(v) = f (v) = t,

h(u) = f (u) − a = s − a.

Properties (ii) and (iii) being self-evident, we show that K2,n is h-choosable. Forthis it suffices to know that h(u)h(v) = (s − a)t > a + b, since vertices xi withf (xi) ≥ 3 are always colorable. But using (6) and the fact that K2,n is f -choosable,we calculate that

(s − a)t = (s − a)(t − a) + a(s − a) > b + a(s − a) ≥ b + a · 1 = a + b.

We can now assume without loss of generality that f (xi) ≥ 2 for i = 1, 2, . . . , n.Suppose there exists an i such that f (xi) > 2, say, f (x1) = · · · = f (xn−m) = 2 andf (xn−m+1), . . . , f (xn) > 2. Let h be defined by

h(xi) = f (xi) = 2, (i = 1, 2, . . . , n − m),

h(xi) = 2 ≤ f (xi) − 1, (i = n − m + 1, . . . , n),

h(v) = f (v),

h(u) = f (u) + m = s + m.

Then∑

v∈V h(v) ≤ ∑v∈V f (v). Since K2,n is f -choosable, st > n − m. Hence

h(u)h(v) = (s + m)t > n − m + mt ≥ n, and hence K2,n is h-choosable. Wenow conclude that we may assume without loss of generality that f (xi) = 2 fori = 1, 2, . . . , n. Since K2,n is f -choosable, we must have st > n. It now follows thatχsc(K2,n) ≥ 2n + α, completing the proof of the theorem. �

Some elementary analysis (which we omit), allows one to determine χsc(K2,n)

explicitly.

Corollary 5. Let n ≥ 1 be an integer. Let m = √n + 1/4 + 1/2 if this is an integer;

otherwise, let m be the closest integer to√

n + 1/4. Then

χsc(K2,n) ={

2n + 2m, if m >√

n,

2n + 2m + 1, otherwise.

Note that for K2,n, the sum of the number of vertices and the number of edges ofK2,n equals 3n + 2. Hence K2,n is not sc-greedy for n ≥ 3; indeed, the gap between3n + 2 and χsc(K2,n) goes to infinity with n.

We also note that Theorem 2 implies that χsc(K2,3) = 10. Removing an edgefrom K2,3 we get a graph G which, using Theorem 1, satisfies χsc(G) = 10. Thus

180 A. Berliner et al.

removing an edge from a graph need not decrease the sum choice number. It isnatural to call a graph H sum choice critical (sc-critical) provided that each graphH ′ obtained from H by removing an edge satisfies χsc(H

′) < χsc(H). Cycles aresc-critical graphs as are complete graphs.

The following theorem may be of some help in determining the sum choice num-ber of a 2-connected graph in that it allows one to impose certain restrictions onthe values of a choice function.

Theorem 3. Let G = (V , E) be a graph, and let u be a vertex of G with degree du. LetG \ u be the graph obtained from G by removing u and its incident edges. Then

χsc(G) ≤ χsc(G \ u) + du + 1 (7)

with equality if and only if there is a minimum choice function f for G such thatf (u) = 1 or f (u) = du + 1.

Proof. Let G \ u be g-choosable. Then it is easy to verify that G is f -choosablewhere f : V → Z+ is defined either by

f (v) ={

g(v), if v �= u,

du + 1, if v = u;

or by

f (v) =

g(v), if v �= u and v is not joined to u,

g(v) + 1, if v is joined to u,

1, if v = u.

In both cases,∑

v∈V

f (v) =∑

v∈V \{u}g(v) + du + 1. (8)

Thus, if g is chosen so that∑

v∈V \u g(v) = χsc(G \ u), then (8) implies that (7)holds; and if equality holds in (7), then (8) implies that

∑

v∈V

f (v) = χsc(G \ u) + du + 1 = χsc(G).

Conversely, if there is a minimum choice function f for G and either f (u) = 1or f (u) = du + 1, then reversing the above definitions to define g in terms off , we see that G \ u is g-choosable and

∑v∈V \u g(v) = χsc(G) − du − 1. Hence

χsc(G \ u) ≤ χsc(G) − du − 1, and by (7) equality occurs. �

We now prove a theorem that will enable us to show that given a positive integern, for each integer e with n − 1 ≤ e ≤ n(n − 1)/2, there exists a connected graph G

with n vertices and e edges such that G is sc-greedy.

Sum List Coloring Graphs 181

Theorem 4. Let n and k be integers with 1 ≤ k ≤ n. Let Gn,k = (V , E) be the graphobtained from the complete graph Kn by adjoining a new vertex u with an edge fromu to each of k different vertices of Kn. Then

χsc(Gn,k) = χsc(Kn) + k + 1 = n(n + 1)

2+ k + 1,

and hence Gn,k is sc-greedy.

Proof. We have χsc(Gn,k) ≤ n(n + 1)/2 + k + 1, since the expression on the rightequals the sum of the number of vertices and the number of edges of Gn,k. Thus weneed to show that Gn,k is not f -choosable for any function f with

∑

v∈V

f (v) ≤ n(n + 1)/2 + k. (9)

So let f be a choice function for Gn,k satisfying (9). Let the vertices of Kn bev1, v2, . . . , vn where f (v1) ≤ f (v2) ≤ · · · ≤ f (vn). We have f (vj ) ≥ j for all j ,since otherwise f (v1), f (v2), . . . , f (vj ) ≤ j − 1 and by choosing color lists forv1, v2, . . . , vj from colors 1, 2, . . . , j − 1, we contradict the assumption that Gn,k

is f -choosable. Since Kn is sc-greedy, we have∑n

i=1 f (vi) ≥ n(n + 1)/2, and henceby (9),

f (u) = a ≤ k andn∑

i=1

f (vi) ≤ n(n + 1)

2+ k − a. (10)

We show that there is an f -family C = (Cv : v ∈ V ) such that Gn,k is not C-listcolorable.

Since the degree of vn is n − 1 or n, we know that f (vn) = n or n + 1. Firstwe assume that f (vn) = n. To each vertex vi we assign the initial lists Cvi

={1, 2, . . . , f (vi)}. Since f (vi) ≥ i for i = 1, 2, . . . , n, these lists determine a (0, 1)-matrix B = [bij ] of order n where bij = 1 if and only if 1 ≤ j ≤ f (vi) for 1 ≤ i ≤ n.Thus B has all 1’s on and below its main diagonal. By (10), there are at most k − a

1’s above the main diagonal of B, and these 1’s are closest to the main diagonal ineach row. There is a unique maximum integer t such that

B =

B1 O O · · · O

J B2 O · · · O

J J B3 · · · O...

......

. . ....

J J J · · · Bt

(11)

where J denotes an all 1’s matrix and the Bi are square matrices of order pi, (i =1, 2, . . . , t). We have

∑ti=1 pi = n and

t∑

i=1

(pi − 1) =t∑

i=1

pi − t ≤ k − a. (12)

182 A. Berliner et al.

The block form (11) of B determines a partition of the vertices {v1, v2, . . . , vn} intosets V1, V2, . . . , Vt where |Vi | = pi, (i = 1, 2, . . . , t).

Recall that the C = (Cvi: i = 1, 2, . . . , n) are initial color lists given by Cvi

={1, 2, . . . , f (vi)}. In any C-list coloring of the vertices v1, v2, . . . , vn of Gn,k, thecolors of the vertices in Vi , (1 ≤ i ≤ t), are collectively determined, that is, the setof colors of the vertices in Vi is uniquely determined as a set Xi of cardinality pi .

Consider the set U of cardinality k consisting of those vertices among v1, v2,

. . . , vn that are joined to u in Gk,n. Let Vi1 , Vi2 , . . . , Vim be the sets from among theVi that are contained in U . Suppose that pi1 +pi2 + · · ·+pim < a. Then using (12),we see that

k = |U | ≤ pi1 + pi2 + · · · + pim +t∑

i=1

(pi − 1) < a + (k − a) = k,

a contradiction. Thus pi1 + pi2 + · · · + pim ≥ a. Choosing as color list Cu a set ofa colors contained in Xi1 ∪ Xi2 ∪ · · · ∪ Xim we conclude that Gn,k does not have a Clist coloring, as desired.

Finally, we assume that f (vn) = n + 1. If u is joined to vn, then we have upondeleting vn, a graph Gn−1,k−1 where, inductively, χsc(Gn−1,k−1) ≥ n(n − 1)/2 + k.Hence

∑

v∈V

f (x) ≥ n(n − 1)

2+ k + n + 1 = n(n + 1)

2+ k + 1.

If u is not joined to vn, then we have upon deleting vn a graph Gn−1,k where in asimilar way we get

∑

v∈V

f (x) >n(n + 1)

2+ k + 1

in contradiction to our assumption that f satisfies (9). �

Corollary 6. Let Tn be a tree on n ≥ 3 vertices. Let m = n(n−1)/2−n+2. Then thereis a sequence G1, G2, . . . , Gm of connected graphs such that G1 = Kn, Gm = Tn,and Gi+1 is obtained from Gi by deleting an edge where χsc(Gi+1) = χsc(Gi) − 1 fori = 1, 2, . . . , m − 1. In particular, each of the graphs G1, G2, . . . , Gm is sc-greedy.

Proof. We prove the corollary by induction on n, it being trivial for n = 3. Assumethat n ≥ 4. Let the vertices of Kn and Tn be labeled v1, v2, . . . , vn where v1 is apendent vertex of Tn with an edge to vertex v2. Let Tn−1 be the tree on n − 1 ver-tices obtained by removing v1 from Tn. Let p = (n − 1)(n − 2)/2 − (n − 1) sothat p + (n − 2) = m. By induction there exists a sequence of p sc-greedy graphsHn−1, Hn, . . . , Hm such that Hn−1 = Kn−1, Hm = Tn−1, and Hj+1 is obtained fromHj by deleting an edge where χsc(Hj+1) = χsc(Hj )− 1 for j = n− 1, n, . . . , m− 1.Let G1 = Kn and G2, . . . , Gn−1 be obtained from G1 by deleting in succession theedges from v1 to v3, v4, . . . , vn. By Theorem 4, G2, . . . , Gn−1 are sc-greedy. Finally,let Gj be obtained from Hj by inserting an edge from v1 to v2 for j = n−1, n, . . . , m.

Sum List Coloring Graphs 183

Then Gm = Tn and using Theorem 1 we conclude that all the Gi are sc-greedy. Thecorollary now follows. �

3. Open Problems

Here we list some open problems for further research. We state some of them quitegenerally.

1. Determine the sum choice number of a 2-connected graph. Less ambitiously,identify certain classes of 2-connected graphs for which the sum choice numberscan be determined.

2. Characterize greedy-sc graphs.3. Characterize sc-critical graphs.4. What is the sum choice number of the graph obtained by joining cycles C4 edge

to edge in a row (the vertex-edge graph of a 1 by n board).5. Let G be a graph properly containing a complete subgraph Kp of size p. Let G

be f -choosable where∑

v∈V f (v) = χsc(G). Is there an f -family C such that inevery C-list coloring of G, the colors of vertices of Kp are uniquely determined(either individually or a a group)? Even the case p = 2 would be of interest. Whatabout ‘forcing’ the colors of any two vertices in the same connected component,joined or not? (cf. Corollary 1)

6. What is the sum choice number of Km,n; in particular, of K3,n?7. What is the sum choice number of the Peterson graph?

We are indebted to two referees for useful comments.

References

1. Erdos, P., Rubin, A.L., Taylor, H.: Choosability in graphs. In: Proceedings of the WestCoast Conference on Combinatorics, Graph Theory and Computing (Humboldt StateUniv., Arcata, Calif., 1979), pp. 125–157, Congress. Numer., XXVI, Utilitas Math., Win-nipeg, Man., 1980

2. Giaro, K., Kubale, M.: Edge-chromatic sum of trees and bounded cyclicity graphs. Inf.Proc. Letters, 75, 65–69 (2000)

3. Isaak, G.: Sum list coloring 2 × n arrays. Elec. J. Combinatorics 9, #N8 (2002)4. Isaak, G.: Sum list coloring block graphs. Graphs and Combinatorics 20, 499–506 (2004)5. Woodall, D.R.: List colourings of graphs. Surveys in Combinatorics. J.W.P. Hirschfeld ed.,

London Math Soc. Lec. Notes Ser. 288, Cambridge Univ. Press, Cambridge, 2001, pp.269–301

Received: June 21, 2004Final version received: June 9, 2005