Extensions of Fractional Precolorings
show Discontinuous Behavior
We study the following problem: given a real number and integer , what is the smallest such that any fractional -precoloring of vertices at pairwise distances at least of a fractionally -colorable graph can be extended to a fractional -coloring of the whole graph? The exact values of were known for and any . We determine the exact values of for if , and if , and give upper bounds for if , and if . Surprisingly, viewed as a function of is discontinuous for all those values of .
1 Introduction and main results
Graph coloring is one of the classical topics in graph theory. In this paper, we seek conditions when a precoloring of some vertices in a graph can be extended to a coloring of the entire graph. This line of research was initiated by Thomassen  who asked for sufficient conditions on extending precolorings of vertices in planar graphs. His original question led to the following result of Albertson .
Theorem 1.1 ().
Let be an -colorable graph and a subset of its vertex set such that the distance between any two vertices of is at least four. Then every -coloring of can be extended to an -coloring of .
It is natural to ask whether an analogue of Theorem 1.1 also holds for non-integer relaxations of colorings. For circular colorings introduced in , the extension problem was almost completely solved by Albertson and West  (see [20, 21] for background and results on circular colorings).
Another well-established relaxation of classical colorings is the notion of fractional colorings, see , which we address in this paper. A fractional -coloring of a graph is an assignment of measurable subsets of the interval to the vertices of such that each vertex receives a subset of measure one and adjacent vertices receive disjoint subsets. The fractional chromatic number of is the infimum over all positive real numbers such that admits a fractional -coloring. For finite graphs (which we restrict our attention to), such exists, the infimum is in fact a minimum, and its value is always rational. A fractional -precoloring is an assignment of measurable subsets of measure one of the interval to some vertices of a graph.
In this paper, we study conditions under which a fractional precoloring can be completed to a fractional coloring of the whole graph.
Let be a real, a rational and an integer. Given a fractionally -colorable graph and a fractional -precoloring of a subset of its vertex set at pairwise distance at least , is it possible to extend the precoloring to a fractional -coloring of the whole graph ?
For a fixed rational and a fixed integer , let be the infimum over all non-negative reals satisfying the following: for any and any fractionally -colorable graph , an arbitrary -precoloring of vertices at pairwise distance at least in can be extended to a fractional -coloring of . The next proposition, which is proved in , implies that for any there exists a fractionally -colorable graph with a fractional -precoloring of some of its vertices at pairwise distance at least , such that there is no extension of the precoloring to a fractional -coloring of .
Proposition 1.2 ().
Let be a graph with fractional chromatic number and a subset of its vertex set. The set of all non-negative reals such that any fractional -precoloring of can be extended to a fractional -coloring of is a closed interval.
Theorem 1.3 ().
For every and , we have:
where . The formula also holds for and .
For we have .
For we have .
For additional values of and , we provide upper bounds (Theorems 3.2, 4.2, 5.2, 6.2, and 6.3) which we believe to be tight. See Figures 2 and 4 for the bounds we can prove for and . To our surprise, for fixed , the function is discontinuous in at , while for the function is also discontinuous at . We provide some additional comments on those observations in Section 7. Also note that the functions and are decreasing on the intervals and , respectively, whereas for all the functions are increasing on .
The paper is organized as follows. In the analysis of the values of , we consider four cases based on the remainder of modulo 4. In Section 3, we present our upper bounds on for and divisible by four. We also present the matching lower bound for . This lower bound is based on a simple expansion bound on independent sets in Kneser graphs based on eigenvalues of its adjacency matrix. In Section 4, we present our upper bounds on for and congruent to two modulo four. This section also contains the matching lower bound for the case and . This lower bound uses a suitable solution of the linear program dual to that for finding the fractional chromatic number of a Kneser graph. Finally, in Sections 5 and 6 we present our upper bounds on for congruent to one and three, respectively.
2 Notation, definitions and preliminary results
Before we can present our results, and their proofs, in detail, we need to introduce some notation. For a positive integer , we set . Next, for a set we write for the set of all measurable subsets of . If is a mapping from a set to and is a subset of , we write for the set . We also write for mappings from to such that for any two distinct .
We gave one possible definition of the fractional chromatic number of a graph in the introduction. An equivalent definition can be given as a linear relaxation of determining the ordinary chromatic number: assign non-negative real weights to the independent sets of such that for every vertex the sum of the weights of independent sets containing is at least one. The minimum possible sum of weights of all independent sets in , where the minimum is taken over all such assignments, is equal to the fractional chromatic number of .
The definition of fractional colorings also allows one to define a class of universal graphs, i.e., a class such that for every graph with fractional chromatic number there is a homomorphism to one of the graphs in this class. A homomorphism from a graph to a graph is a mapping such that if and are two adjacent vertices of , then the vertices and are adjacent in . If such a mapping exists, we say that is homomorphic to .
Universal graphs for fractional colorings are Kneser graphs ; the graph , for integers , has a vertex set formed by all -element subsets of , i.e., . Two vertices and are adjacent if . It is not hard to show that the fractional chromatic number of is equal to . The following proposition can be found, e.g., in .
Let be a graph with fractional chromatic number . There exist integers and such that and is homomorphic to the graph .
Analogously to , our proofs are based on defining and analyzing graphs that are universal for graphs (of a given fractional chromatic number) with some precolored vertices. The graphs we introduce now are isomorphic to the ones defined in , although we use a slightly different notation.
The extension product of two graphs and is the graph with vertex set such that vertices and are adjacent if and are adjacent in and either , or and are adjacent in . This type of a graph product was introduced by Albertson and West . An equivalent notion was used in  under the name universal product; the only difference is that the meaning of and was swapped, i.e., the universal product of and is isomorphic to the extension product of and . For a set , a ray is the extension product of the Kneser graph and the -vertex path with vertices ; the vertex of is marked as special. The copy of in the ray corresponding to the vertex of the path is said to be the base of the ray. For brevity, will stand for in what follows. The ray is sketched in Figure 5. Note that the graph is homomorphic to , and the distance between the vertex and any vertex , for and , is at least .
The graph , which we now define, is a universal graph for graphs with fractional chromatic number with precolored vertices at pairwise distance at least . Fix positive integers and such that and . If is even, the graph is the extension product of the Kneser graph and the star with each edge subdivided times. For every , we mark the vertex as special in copies of corresponding to the leaves of the star (for different values of , we choose different copies). In this way, the subgraphs of corresponding to the products of the subdivided edges and are isomorphic to rays . Hence, the graph can be viewed as obtained from copies of the ray for each choice of through identification of the bases of the rays. The graph is sketched in Figure 6.
For positive integers and , let be the graph obtained from a clique by identifying each vertex of the clique with an end-vertex of a -vertex path; so , for , has vertices of degree two, vertices of degree one, and vertices of degree . If is odd, the graph is the extension product of the Kneser graph and the graph . Again, for each , we mark vertices in of the copies of corresponding to the vertices of degree one of as special (with different copies for different values of again). In this way, we can view as a union of rays with additional edges between their bases. The graph is sketched in Figure 7.
In the next three propositions, we summarize the properties of the graphs needed in the proofs. We start with the first two of them; the proof of the first one is straightforward and the proof of the second one is in .
The graph for and is homomorphic to and its special vertices are at pairwise distance at least .
Proposition 2.3 ().
Let be a graph with fractional chromatic number and a subset of its vertex set at pairwise distance at least . There exist positive integers and , such that and the graph has a homomorphism to that maps the vertices of to distinct special vertices of .
The length of the shortest odd cycle of a graph is the odd girth of . The odd girth of the Kneser graph is equal to , see . Note that Proposition 2.1 implies that if is a fractionally -colorable graph, then its odd girth is at least . The main difference between the case , which was fully analyzed in , and the case is that vertices of a ray at some fixed small distance from the special vertex form an independent set. Observe that the minimum distance for which this property does not hold is related to the odd girth of the Kneser graph .
Consider a special vertex of a universal graph and an integer . The vertices at distance from form an independent set in .
Proposition 2.5 ().
Let be rational, where and , and . For every fractional -precoloring of the special vertices of by subsets there exist functions and from to such that the following holds:
for every , : and ;
for every and :
In other words, the function in Proposition 2.5 is an equipartition of the interval into measurable parts such that the measure of the intersection of with each set , for and , is the same as the expected intersection of with a random subset of of measure . Analogously, is a partition of an appropriate subset of of measure into measurable parts , where the parts have measure and the measure of the intersection of with each set is the same as for a random subset of of measure .
3 Distances divisible by four
3.1 Upper bounds
In this section we prove upper bounds on for in the case that and satisfy . Observe that Proposition 2.4 guarantees that if we consider the ray , then for any , the vertices at distance from the special vertex form an independent set.
Let be a positive real and , , and positive integers such that and . If the conditions
are satisfied, where and , then any fractional -precoloring of the special vertices of can be extended to a fractional -coloring of .
Furthermore, it is straightforward to show that any positive solution to this equation satisfies the following two inequalities as well:
These two inequalities will guarantee the existence of functions and , respectively, which we define later in the proof. Also note that the right inequality of (3) is an immediate consequence of the left one.
Now consider the universal graph . Let , for , be a precoloring of the special vertices and let be a mapping as described in Proposition 2.5. In what follows, for each ray , which is isomorphic to , we find a fractional coloring that satisfies the following: for every set , each vertex of the base of is colored by the set , and the special vertex of is colored by . Since the universal graph is constructed by identifying the vertices , the conclusion of the lemma follows from the existence of such a fractional coloring for each ray.
Fix a ray and let be the special vertex of . For an integer , let be the set of vertices of at distance from , and let be the set of vertices of at distance at least from . Observe that the sets , , form a partition of , and if a vertex of the ray is in , then . In particular, the vertices of the base of form a subset of . By (1) and Proposition 2.4, it follows that the set forms an independent set in , for .
The basic idea is to partition for each the interval into three parts. The first part will be split into equal-size parts and will be assigned to vertices in according to the corresponding sets in the Kneser graph. The second part will be assigned to all vertices in (that is possible since forms an independent set). The third part will not be used on the vertices of at all and will be reserved for the vertices in . Based on the parity of , either the second part will be inside and the third part will be disjoint from , or vice versa. First we define the partition for , and after defining the partition for some , we define the partition for . During this procedure, the sizes of the second and third parts will increase at the expense of the first part.
Formally, we construct functions , and , for , and in the following way. For and , we sequentially define:
as an arbitrary subset of
of measure ,
as an arbitrary subset of
of measure ,
Finally, we set for every . Since the measure of is and the measure of is , these functions exist if and only if the conditions (3) are satisfied. Next, we set to be the set of measure that is disjoint from , i.e., . Observe that . The described construction of the functions is sketched in Figure 8.
Let and . Recall that . If is even, we set
if is odd, we set
and for we set
Finally, we set .
We claim that for every vertex . Indeed, if , then the assertion immediately follows from . Hence, in the remainder we may assume that belongs to a set for some . Observe that for a fixed , the color sets of any two vertices and from have the same measure. Let be the measure of vertices in . Then , by the definition of . If , then , since both and , for , have measure . Next, if , then
Analogously, if , then
Finally, for we have
which is at least one by (2).
It remains to check that the mapping assigns disjoint sets to any two adjacent vertices in . Let and be two arbitrary adjacent vertices in . Hence, is disjoint from and without loss of generality . If , then (since for the set is independent). Thus the sets and are disjoint, since and are disjoint.
From now on, we assume that . If is even, then is disjoint from , and disjoint from for any . Furthermore, is disjoint from for any . Analogously if is odd and larger than one, then is disjoint from for any , and is disjoint from , and disjoint from for any . Since is a subset of for any and , the sets and are disjoint. Finally, the sets assigned to neighbors of are disjoint from .
We can conclude that the coloring is a fractional coloring of the ray with the required properties. ∎
Let be a positive integer such that , a rational and a positive real such that conditions (1) and (2) are satisfied, where . If is a fractionally -colorable graph and is a subset of its vertex set with pairwise distance at least , then any fractional -precoloring of can be extended to a fractional -coloring of .
Let and be integers such that , and the homomorphism from to given by Proposition 2.3. Precolor the vertices of with the colors assigned to their preimages. Note that this is possible since restricted to is injective. Since the parameters , and satisfy the conditions (1) and (2), Lemma 3.1 yields that there exists an extension of this precoloring of to a fractional -coloring of . Since is a homomorphism of to , setting for all yields a fractional -coloring of that extends the given precoloring of . ∎
3.2 Lower bound for distance four
We start this section with the following proposition about the size of the neighborhood of an independent set in a Kneser graph.
Let and be positive integers, . If is an independent set of the Kneser graph , then .
Let and be the normalized adjacency matrix of the Kneser graph . This is the matrix indexed by vertices of such that if is an edge of , the entry corresponding to is equal to the inverse of the degree of , i.e., equal to , while all other entries are zero. If are the eigenvalues of such that , then it follows that , see .
A standard expansion inequality (see, e.g., [11, Theorem 4.15]) asserts that
for every vertex subset of a graph of size at most , where is the number of vertices of . If is an independent set of the Kneser graph , then by the Erdős-Ko-Rado Theorem (see, e.g., ), the size of is at most . Therefore, , where , and hence by (4). ∎
Proposition 3.3 has a key role in proving that in any fractional -coloring of , where , there is a vertex such that the union of sets assigned to the neighborhood of has measure at least . Note that this statement is trivial if , because in that case the neighborhood of any vertex of is isomorphic to .
For every real , all positive integers and , where , and any fractional -coloring of , there exists a vertex such that .
For , let be the set of vertices of that contain in their color set, i.e., . For we define . In other words, are the points in contained in exactly color sets . Note that for the set is empty and that . Next, let be the set of points such that the number of vertices that have at least one neighbor with is equal to . In other words, let .
Finally, consider all the intersections of with , where and , and let . Note that for a fixed the sets form a partition of the set , where for some values of the part might be empty. Since for any the set forms an independent set in , Proposition 3.3 yields that if , then is empty. Now, for a vertex , consider the measure of points such that is contained in the color set of at least one neighbor of . By a double counting argument it follows that
Since the sets are empty for , we conclude that
Therefore, there exists a vertex such that . ∎
We are now ready to prove that the upper bound on for given in Theorem 3.2 is best possible. The proof uses the same precoloring as was used in  for a lower bound in the case , but the argument for is considerably more involved.
Let be a rational and a positive real such that . There exist a graph with fractional chromatic number , a subset of its vertex set at pairwise distance at least four and a fractional -precoloring of that cannot be extended to a fractional -coloring of .
Let be the positive root of the equation