1 Introduction
A coloring of a a graph is a map . A coloring is proper if no two adjacent vertices are assigned the same color. If there is a proper coloring of a graph that uses at most colors then we say that is colorable and that is a coloring for . The coloring problem Col asks, for a given graph and a positive integer , whether there is a coloring for . When is fixed, we have the Coloring problem.
A list assignment is a map assigning a set of positive integers to each vertex of . Given and , the List Coloring problem LiCol asks for the existence of a proper coloring that obeys , i.e., each vertex receives a color from its own list. If the answer is positive, is said to be colorable. Variants of the problem are defined by bounding the total number of available colors or by bounding the list size. In List Coloring (Li Col), for each . Thus, there are colors in total. On the other hand in List Coloring (LiCol) each list has size at most , in this case the total number of colors can be larger than .
Precoloring Extension, PrExt, is a special case of LiCol and a generalization of Col. In PrExt all of the vertices in a subset of are previously colored and the task is to extend this coloring to all of the vertices. If, in addition, the total number of colors is bounded, say by , then it is called the kPrecoloring Extension, PrExt. Col is clearly a special case of PrExt, which in turn is a special case of Li Col. Refer to golovach2016 for a chart summarizing these relationships.
For general graphs Col and its variants LiCol and PrExt are NP–complete; see karp1972 ; garey1979 . Most of their variants are NPcomplete even when the parameter is fixed for small values of : Col, LiCol, Li Col and PrExt are NPcomplete when lovasz1973 and they are polynomially solvable when erdos1979 ; vizing1976 .
Concerning the complexity of these problems in graph classes, Col is solvable in polynomial time for perfect graphs grostchel whereas LiCol is NPcomplete when restricted to perfect graphs and many of its subclasses, such as split graphs, bipartite graphs kubale1992 and interval graphs biro1992 . On the other hand LiCol is polynomially solvable for trees, complete graphs and graphs of bounded treewidth jansen1997 . Refer to Tuza tuza1997 and more recently to Paulusma paulasma2016 for related surveys.
For small values of , Jansen and Scheffler jansen1997 have shown that –LiCol is NPcomplete when restricted to complete bipartite graphs and cographs, as observed in golovach2014 . Kratochvíl and Tuza kratochvil1994 showed that –LiCol is NPcomplete even if each color appears in at most three lists, each vertex in the graph has degree at most three and the graph is planar. PrExt is NPcomplete even for –regular planar bipartite graphs and for planar bipartite graphs with maximum degree chlebik .
For fixed , Li Col is polynomially solvable for free graphs hoang2010 . Note that chordal bipartite graphs contain free graphs but free graphs are incomparable with chordal bipartite graphs spinrad . Li Col is polynomial for free graphs broersma2013 and for free graphs bonomo2018 . Computational complexity of Li Col for free bipartite graphs is open bonomo2018 . Even the restricted case of Li Col for free chordal bipartite graphs is open. Golovach et. al. golovach2016 give a survey that summarizes the results for Li Col on free graphs in terms of the structure of .
PrExt problem is solvable in linear time on free graphs and it is NPcomplete when restricted to free chordal bipartite graphs hujter1996 . PrExt is NPcomplete even for planar bipartite graphs kratochvil1993 , even for those having maximum degree 4 chlebik . Recall that PrExt generalizes PrExt and Li Col generalizes PrExt. But there is no direct relation between PrExt and Li Col golovach2016 .
Coloring problems can be placed in the more general class of –coloring problems. Given two graphs and , a function such that and are adjacent in whenever and are adjacent in is called a graph homomorphism from to . For a fixed graph and for an input , the coloring problem, Col, asks whether there is a to homomorphism. In the list coloring problem, Li Col, each vertex of the input graph is associated with a list of vertices of and the question is whether a to homomorphism exists that maps each vertex to a member of its list. Observe that Li Col is a generalization of Li Col. The complexities of the –coloring and list –coloring problems for arbitrary input graphs are completely characterized in terms of the structure of , see Nešetřil and Hell nesetril .
Although intensive research on this subject has been undertaken in the last two decades, there are still numerous open questions regarding computational complexities on LiCol and its variants when they are restricted to certain graph classes. Huang, Johnson and Paulusma huang2015 proved that Li Col is NPcomplete for –free chordal bipartite graphs and –PrExt is NPcomplete for –free chordal bipartite graphs. They further pose the problem on the computational complexity of the Li Col and PrExt on chordal bipartite graphs. Here Li Col and PrExt on convex bipartite graphs, a proper subclass of chordal bipartite graphs, are studied and a partial answer to this question is given. Figure 1 summarizes the related results.
A bipartite graph is convex if it admits an ordering on one of the parts of the bipartition, say , such that the neighbours of each vertex in are consecutive in this order. If both color classes admit such an ordering the graph is called biconvex bipartite (see Section 2 for formal definitions). Chordal bipartite graphs contain convex bipartite graphs properly. Convex bipartite graphs contain as a proper subclass biconvex bipartite graphs which contain bipartite permutation graphs properly. More information on these classes can be found in Spinrad spinrad and in Brandstädt, Le and Spinrad brandstadt .
Enright, Stewart and Tardos enright2014 have shown that Li Col is solvable in polynomial time when restricted to graphs with all connected induced subgraphs having a multichain ordering. They apply this result to permutation graphs and interval graphs. Here we show that connected biconvex graphs also admit a multichain ordering, implying a polynomial time algorithm for Li Col on this graph class.
From the point of view of parameterized complexity, treewidth can be computed in polynomial time on chordal bipartite graphs kloks1993 . Li Col can be solved in polynomial time on chordal bipartite graphs with bounded treewidth jansen1997 ; diaz2002 which includes chordal bipartite graphs of bounded degreee lozin . Li Col is polynomial for graphs of bounded cliquewidth courcelle . Note that convex bipartite graph contains graphs with unbounded treewidth as well as graphs with unbounded cliquewidth.
The paper is organized as follows. In Section 2 we give the necessary definitions. In Section 3 we show that connected biconvex bipartite graphs admit multichain ordering. In Section 4, we show that Li Col is polynomially solvable when it is restricted to convex bipartite graphs. Then we show how to extend this result to Li Col.
2 Preliminaries
We consider finite simple graphs . For terminology refer to Diestel diestel .
An edge joining non adjacent vertices in the cycle, , is called a chord. A graph is chordal if every induced cycle of length has a chord. Chordal bipartite graphs are bipartite graphs in which every induced has a chord. This graph class is introduced by Golumbic and Gross golumbic1978 . Chordal bipartite graphs may contain induced , so they do not constitute a subclass of chordal graphs but it is a proper subclass of bipartite graphs. Chordal bipartite graphs can be recognized in polynomial time paige1987 .
A bipartite graph is represented by , where , form a bipartition of the vertex set into stable sets. An ordering of the vertices in a bipartite graph has the adjacency property (or the ordering is said to be convex) and is said to have convexity with respect to if, for each vertex , consists of vertices which are consecutive in the ordering of . We say that an ordering of the vertices in a bipartite graph has the enclosure property if for every pair of vertices such that , the vertices in occur consecutively in the ordering of
Convex bipartite graphs are bipartite graphs that have the adjacency property on one of the partite sets and biconvex bipartite graphs have the the adjacency property on both partite sets and . Fig. 2 shows a graph that is convex but not biconvex. Bipartite permutation graphs are biconvex bipartite graphs in which one of the partite sets obeys both the adjacency and the enclosure properties. There are linear time recognition algorithms for these classes spinrad1987 ; nussbaum2010 .
A chain graph is a bipartite graph that contains no induced (a graph formed by two independent edges) yannakakis1982 . The following characterization from enright2014 is equivalent: a connected bipartite graph with bipartite sets and is a chain graph if and only if for any two vertices we have or . If the vertices in are ordered with respect to their degrees starting from the highest degree, then for any , the vertices in will be consecutive in the ordering on and, if the graph is connected, there is always a vertex so that includes the first vertex in . In particular, chain graphs are a proper subclass of convex bipartite graphs.
3 List Coloring on Biconvex Graphs
Enright, Stewart and Tardos enright2014 show that Li Col, as well as the general Li Col, is solvable in polynomial time when restricted to graphs with all connected induced subgraphs having a multichain ordering. They apply this result to permutation graphs and interval graphs. Here we show that connected biconvex graphs also admit a multichain ordering.
The distance layers of a connected graph from a vertex are where and, for , consists of the vertices at distance from and is the largest integer for which this set is nonempty (see Figure 3 for an example). These layers form a multichain ordering brandstadt2003 of if, for every two consecutive layers and , the edges connecting these two layers form a chain graph (not necessarily the layers themselves). All connected bipartite permutation graphs brandstadt2003 and interval graphs enright2014 admit multichain orderings. Observe that, for the graph given in Fig. 3, the distance layers from provide a multichain ordering.
Recall that a subdivision of a graph is the graph obtained from by replacing each edge by a path of length two. Thus and .
Lemma 1
If is a biconvex graph, then does not contain as an induced subgraph.
Proof
Let be a biconvex graph and let . Let be the vertex of degree in , and be the vertices in and and the vertices of degree so that is adjacent to for , see Fig. 4.
We observe that there is no ordering of in which the three sets , and become consecutive. Therefore, a bipartite graph which contains as an induced subgraph does not admit a biconvex ordering.
Proposition 1
Every connected biconvex graph admits a multichain ordering.
Proof
To see that biconvex graphs admit a multichain ordering, we use the notion of biconvex straight ordering introduced by Abbas and Stewart stewart2000 . Let be a bipartite graph with a linear ordering defined on . Two edges , where and , are said to cross if and . If and cross, we call and the corresponding straight pairs. An ordering on is a straight ordering if, for each pair of crossing edges, at least one of the corresponding straight pairs, or , is an edge of the graph stewart2000 .
Let be a connected biconvex graph. It follows from (stewart2000, , Theorem 11) that admits a biconvex straight ordering, say of . Let be the distance layers of from . Since the graph is connected, . Let us show that these layers form a multi–chain ordering.
The first layers and trivially form a multi–chain ordering. Let , where the vertices are listed according to the ordering. When , trivially form a chain graph. When , since the ordering is straight, all the edges joining with vertices in cross with the edge . As, is not connected to , the other straight pair should be an edge in . Therefore . By iterating the same argument, we see that . Thus the layers form a multi–chain ordering.
When , let us show that form a multichain ordering. When , trivially form a multichain ordering. Otherwise, assume that and, for a contradiction, that the bipartite graph induced by contains an induced copy of , say with edges with and . Since the ordering is straight, we may assume that and . Since , and the ordering is biconvex, we must have . But then contains but not , contradicting the biconvexity of the ordering. Thus form a multi–chain ordering.
Suppose that . Let be the largest subscript such that form a multichain ordering. Suppose for a contradiction that , Thus the bipartite graph induced by the layers contain an induced copy of , say with edges , and . As the ordering is straight, we may assume and . We consider two cases:
Case 1: . Let and consider predecessors of . Then the subgraph induced by is isomorphic to a subdivision of , contradicting Lemma 1.
Case 2: . Let and be some predecessors of and in the previous layer. Observe that the two edges induce a in the subgraph induced by contradicting the choice of .
Proposition 1 and the main result by Enright, Stewart and Tardos (enright2014, , Theorem 2.1) give us our main result in this section.
Theorem 3.1
For any , Li Col is solvable in polynomial time when restricted to biconvex graphs.
As Li Col is a particular case of Li Col and Li Col generalizes PrExt, we have the following corollary.
Corollary 1
Li Col and PrExt are solvable in polynomial time when restricted to biconvex graphs.
Concerning the running time of the algorithms, it is shown in Abbas and Stewart stewart2000 that a biconvex straight ordering of a biconvex bipartite graph can be found in linear time on the number of vertices of the graph. On the other hand, the algorithm in enright2014 is shown to run in time time when a multichain ordering in decreasing ordering of degrees is given. Observe that to get such ordering we have only to reorder the elements in the layers provided by the straight ordering, therefore it can be obtained in linear time. All together gives an upper bound on the complexity of Li Col in the class of biconvex graphs.
4 List Coloring of Convex Bipartite Graphs
Let be a connected bipartite graph that is convex with respect to . Let be a convex ordering of , that is, for each there are two positive integers such that
Consider the set of integers and . For the graph given in Fig. 5, and .
We use the set to direct the dynamic programming algorithm and the elements in to determine the relevant information to be kept for the next step. Assume that are sorted so that . By connectivity of , we have . For each , let , , and . Define . Observe that , and that contains those vertices in whose neighborhood starts before or at and ends after . For example, for the graph given in Fig. 5, and . For sake of simplicity we assume an initial point , so that is the empty graph.
Let be a set of colors. Assume that each vertex in has an associated list . We next define the information that we want to compute for each . For each , define . As before we assume that the elements in are increasingly ordered, . To simplify notation set to make sure that a higher value always exists. For the example in Fig. 5, for instance, . For the fictitious initial value , we take
Fix , . For each and , will hold value true whenever there is a valid list coloring of such that it uses no color in for the set . Observe that we are not considering as a potential set as not using any color is impossible.
The Color Algorithm will compute those values in three steps. In going from to , first it computes the values for the that were not in combining this information with the relevant information computed in the previous step. Next, it incorporates the restriction from that were not in . Finally, it rearranges the information to keep only the values for the index in .
 Color Algorithm:

Let , . Initially set , and set to TRUE for any . When assume that the values of have already been computed.
 Step 1

Extending to new parts.
Let . For , by construction, those values lie before and some of them have no corresponding entries in . Assume that increasingly ordered. Let . We set for and to be true whenever there is a valid list coloring of the set . For this, the algorithm checks whether for each . If this is the case, one can select a color not in and get a valid coloring. Accordingly we update the value of so that it remains TRUE if it was already set to TRUE and the previous condition holds for the elements in
 Step 2

Incorporating .
For and , consider any entry set to TRUE. If , the corresponding entry is changed to FALSE.
Next, the values on are processed in increasing order of : any entry holding value TRUE will remain TRUE whenever there is an entry holding value TRUE with . By monotonicity, the property holds whenever is TRUE.
After processing , if holds true, for each piece between and , we can pick a common color not in but in to color that is compatible with some list coloring on the relevant parts that do not use .
 Step 3

Compacting to get .
For each the set might contain several subintervals on , considered either in that will not be needed later on. We fusion those sets from left to right, adding one at a time, setting to true whenever there are corresponding entries holding value true for sets and so that .
To examplify the Color Algorithm consider the list assignment for the graph given in Fig. 5. In the Tables 1 and 2 below the value is calculated for each subinterval in and for each nonempty proper subset of .
T  T  T  F  F  T  
T  T  T  F  T  T  
T  T  T  F  T  T 
T  T  F  F  F  T  
T  T  F  F  T  T  
T  T  F  F  T  T 
Lemma 2
Let be a connected convex bipartite graph, be a color assignment for . There is an –coloring of if and only if there is such that at the end of the execution of the Color Algorithm .
Proof
Assume that admits a list coloring. Let be an coloring of . For let . Observe does not use any color in on and furthermore, for any so that , . Using this fact it follows that the entries in the tables for the corresponding sets get the value true and at the end of the algorithm will be true.
Conversely, we can prove that the Color Algorithm correctly computes the values of for . The proof is by induction. Observe that for the table provides the right indices and the initialization step provides the correct values for the table on an empty graph. By induction hypothesis, we assume that the values of are correctly computed. Step 1 guarantees that the desired coloring exists when adding only the part on to . Step 2, has two parts. The first one guarantees that only those entries with sets that are compatible with the list of the vertices in are still alive. The second one ensures that when combining two consecutive pieces having a common neighborhood on a common set of colors (a subset) is available to color these vertices. Finally Step 3, merge tables for pieces that have the same neighborhood outside , again we need to maintain a common set of colors free for potential use on this neighbors.
Finally observe that all the running time of the color algorithm is polynomial in and in . Furthemore, the PrExt can be polynomially reduced to Li Col. Therefore we get our main result.
Theorem 4.1
For , Li Col and PrExt on convex bipartite graphs can be solved in polynomial time.
The color algorithm can be modified to solve the Li Col on convex bipartite graphs. For this, the algorithm keeps track instead of the unused color on the part of the used ones. For doing that, we have to consider some longer subdivision of the intervals in the part. Step 2 will check that at least one of the colors in the list of is connected to all the used colors in the part. Step 3 is also modified as the global set of used colors will be the union.
Theorem 4.2
For any , Li Col on convex bipartite graphs can be solved in polynomial time.
5 Conclusions
In this paper the problem posed by Huang et al. huang2015 on the computational complexity of the Li Col and PrExt on chordal bipartite graphs is addressed. A partial answer to a general version of this question is given by increasing the subclasses of chordal bipartite graphs for which polynomial time algorithms for the Li Col are known to biconvex bipartite graphs and convex bipartite graphs. Note that the later class includes convex bipartite graphs with bounded degree, complete bipartite graphs which have unbounded treewidth, as well as graphs with unbounded cliquewidth. Interestingly enough the second result can also be extended, with a slight modification, to solve Li Col for the same graph class. The paper includes another result of independent interest: any connected biconvex bipartite graph admit a multichain ordering.
On the other hand, chordal bipartite graphs form a much larger graph class. Using the terminology of scheinerman1994 it is a superfactorial graph class whereas convex bipartite graphs is a factorial graph class. Although Li Col is hard for when restricted to chordal bipartite graphs, finding the computational complexity of Li Col for chordal bipartite graphs is the next natural open question.
Acknowledgments
J. Díaz and M. Serna are partially supported by funds from MINECO and EU FEDER under grant TIN201786727C21R) AGAUR project ALBCOM (2017SGR786) Öznur Yaşar Diner is partially supported by the Scientific and Technological Research Council of Turkey (TUBITAK) BIDEB 2219 [grant number 1059B191802095] and by the Kadir Has University BAP [grant number 2018BAP08]. Oriol Serra is supported by the Spanish Ministry of Science under project MTM201782166P.
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