1.1. Poset Dimension and Degree
The dimension of a poset , denoted by , is the minimum number of total orders whose intersection is . Let be the maximum dimension of a poset whose comparability graph has maximum degree at most . Several bounds on have been proved in the literature. In unpublished work referenced in [44, 25], Rödl and Trotter proved the upper bound, . Füredi and Kahn  improved this result to
On the other hand, Erdős, Kierstead, and Trotter  proved the lower bound,
Both these proofs use probabilistic methods. The problem of narrowing the gap between (1) and (2) was described as “an important topic for further research” by Erdős et al. ; Trotter  speculated that the lower bound could be improved and wrote “a really new idea will be necessary to improve the upper bound—if this is at all possible”; and Wang [47, page 52]
described the problem as “one of the most challenging (and probably quite difficult) problems in dimension theory.”
1.2. Boxicity and Degree
We prove (3) via the notion of boxicity. The boxicity of a (finite undirected) graph , denoted by , is the minimum integer , such that is the intersection graph of boxes in . Here a box is a Cartesian product , where is an interval for each . So a graph has boxicity at most if and only if there is a set of -dimensional boxes such that if and only if . Note that a graph has boxicity 1 if and only if it is an interval graph. It is easily seen that every graph has finite boxicity.
Let be the maximum boxicity of a graph with maximum degree . It is easily seen that , and Adiga and Chandran  proved that . Chandran, Francis, and Sivadasan  proved the first general upper bound of , which was improved to by Esperet . A breakthrough was made by Adiga, Bhowmick, and Chandran  via the following connection to poset dimension.
Given a graph , let be the poset on where if and only if and , and or . Adiga et al.  proved that . The comparability graph of has maximum degree . Thus . Conversely, Adiga et al.  proved that if is the comparability graph of a poset , then , implying . This says that . Thus Adiga et al.  concluded from (1) and (2) that
We improve the upper bound, giving the following result, which is equivalent to (3):
Again, a more precise result is given below (see Theorem 12).
1.3. Boxicity and Genus
Now consider the boxicity of graphs embeddable in a given surface. Scheinerman  proved that every outerplanar graph has boxicity at most 2. Thomassen  proved that every planar graph has boxicity at most 3 (generalised to ‘cubicity’ by Felsner and Francis ). Esperet and Joret  proved that every toroidal graph has boxicity at most 7, improved to 6 by Esperet . The Euler genus of an orientable surface with handles is . The Euler genus of a non-orientable surface with cross-caps is . The Euler genus of a graph is the minimum Euler genus of a surface in which embeds (with no crossings). Esperet and Joret  proved that every graph with Euler genus has boxicity at most . Esperet  improved this upper bound to and also noted that there are graphs of Euler genus with boxicity , which follows from the result of Erdős et al.  mentioned above. See  for more on the boxicity of graphs embedded in surfaces.
1.4. Boxicity and Layered Treewidth
The third contribution of the paper is to prove a new upper bound on boxicity in terms of layered treewidth, which is a graph parameter recently introduced by Dujmović et al.  (see Section 5). This generalises the known bound in terms of treewidth, and leads to generalisations of known results for graphs embedded in surfaces where each edge is in a bounded number of crossings.
1.5. Related Work
The present paper can be considered to be part of a body of research connecting poset dimension and graph structure theory. Several recent papers [46, 33, 31, 28, 29, 30, 32, 35, 42] show that structural properties of the cover graph of a poset lead to bounds on its dimension. Finally, we mention the following relationships between boxicity and chromatic number. Graphs with boxicity 1 (interval graphs) are perfect. Asplund and Grünbaum  proved that graphs with boxicity 2 are -bounded. But Burling  constructed triangle-free graphs with boxicity 3 and unbounded chromatic number.
Roberts  introduced boxicity and proved the following two fundamental results.
Lemma 1 ().
For all graphs such that ,
Note that Lemma 1 is proved trivially via the product construction.
Lemma 2 ().
Every -vertex graph has boxicity at most .
A graph is -degenerate if every subgraph has a vertex of degree at most . Note that -degenerate graphs (that is, forests) have boxicity at most 2, but -degenerate graphs have unbounded boxicity, since the 1-subdivision of is -degenerate and has boxicity . Adiga et al.  proved the following bound. Throughout this paper, all logarithms are natural unless otherwise indicated.
Lemma 3 ().
Every -degenerate graph on vertices has boxicity at most .
The following lemma, due to Esperet , is the starting point for our work on embedded graphs.
Lemma 4 ().
Every graph with Euler genus has a set of at most vertices such that has boxicity at most 5.
Let . For our purposes, a permutation of a set is a bijection from to . A set of permutations of a set is -suitable if for every -subset of and for every element , there is permutation such that for all . This definition was introduced by Dushnik ; see [41, 13, 8] for further results on suitable sets. Spencer  attributes the following result to Hajnal. We include the proof for completeness, and so that the dependence on is absolutely clear (since Spencer assumed that is fixed).
Lemma 5 ().
For every and there is a -suitable collection of permutations of size at most .
A sequence of subsets of is -scrambling if for every set with and every , we have
For , let be the maximum cardinality of a -scrambling family of subsets of . Note that is monotone increasing in (and trivially ).
Let . If is a positive integer such that
Proof of Claim..
Choose subsets of independently and uniformly at random. For any -set and any , the probability that (7) is not satisfied is . There are choices for and then choices for , so taking a union bound, the probability that there is some pair such that (7) is not satisfed is at most the left hand side of (8) (note that it is enough to consider sets of size exactly ). Since this is smaller than 1, we are done. ∎
Given and , choose minimal so that . Let , let be a -scrambling set of subsets of , and let be distinct subsets of . We define orders on as follows: for , let ; then if either
and ; or
(Note that if then this gives the lex order on the , and if it is reverse lex.)
Now is a -suitable collection of orders on . This is straightforward, but a little tricky: given a set of elements of and , let , let , and choose an element of the intersection on the left hand side of (7). Consider the order . It is enough to show that for each . Given , let . If then and so , and therefore ; if then and so , and again .
We will also use the Lovász Local Lemma:
Lemma 6 ().
Let be events in a probability space, each with probability at most and mutually independent of all but at most other events. If then with positive probability, none of occur.
For a graph and set , the graph with vertex set and edge set is called the subgraph of induced by . Let be the graph obtained from by adding an edge between every pair of non-adjacent vertices at least one of which is not in .
Given a -dimensional box-representation of , delete the boxes representing the vertices in to obtain a -dimensional box-representation of . Thus . Given a -dimensional box-representation of , for every vertex in , add a box with interval in every dimension (so that it meets all other boxes). We obtain a -dimensional box-representation of . Thus . ∎
For a graph and disjoint sets , the graph with vertex set and edge set is called the bipartite subgraph of induced by . For non-adjacent vertices and , we say is a non-edge of . Let be the graph obtained from by adding an edge between distinct vertices and whenever or .
3. Bounded Degree
The first ingredient in our proof is the following colouring result that bounds the number of monochromatic neighbours of each vertex. A very similar result was proved by Hind et al. ; they required the additional property that the colouring is proper, but had , which is too much for our purposes.
For every graph with maximum degree and for all integers and , there is a partition of , such that for each and .
Colour each vertex of independently and randomly with one of colours. Let be the corresponding colour classes. For each set of exactly vertices in , such that for some vertex , introduce an event which holds if only if for some . Each such event has probability . The colour on one vertex affects at most events. Thus each event is mutually independent of all but at most other events, where
It follows that . By Lemma 6, with positive probability, no event occurs. Thus the desired partition exists. ∎
Note that an example in , due to Noga Alon, shows that the value of in Lemma 8 is within a constant factor of optimal. Lemma 8 leads to our next lemma. A similar result was used by Füredi and Kahn  in their work on poset dimension.
For every graph with maximum degree and for all integers and , there is a partition of , such that for each and .
Since , we have and and . Thus . The result follows from Lemma 8. ∎
The next lemma is a key new idea in our proof. Its proof is a straightforward application of the Lovász Local Lemma.
Let be a bipartite graph with bipartition , where vertices in have degree at most and vertices in have degree at most . Let be positive integers such that
Then there exist colourings of , each with colours, such that for each vertex , for some colouring , each colour is assigned to at most neighbours of under .
For and for each vertex , let be a random colour in . Let be the event that for each , some set of neighbours of are monochromatic under . The probability that there is a monochromatic set of at least neighbours of under is
Thus . Observe that is mutually independent of all but at most other events. By assumption, . By Lemma 6, with positive probability no event occurs. Therefore, the desired colourings exists. ∎
Let be a bipartite graph with bipartition , where vertices in have degree at most and vertices in have degree at most , for some . Let be the graph obtained from by adding a clique on and a clique on . Then, as ,
Let and and . As , we may assume that is large.
By Lemma 10, there exist colourings of , each with colours, such that for each vertex , for some colouring , each colour is assigned to at most neighbours of . Let be a partition of such that for each , at most neighbours of are assigned the same colour under . Assume that is the set of colours used by each .
Our aim is to construct a box representation for each of the graphs , and then take their intersection using Lemma 1. In light of Lemma 7, it is enough to concentrate on the subgraphs . To handle , we further decompose according to as follows. For each and each colour , let . Let . Note that .
We now bound the boxicity of . Let be the graph with vertex set , where distinct vertices are adjacent in whenever and have a common neighbour in . Since each vertex in has at most neighbours in , the graph has maximum degree at most . Thus . Let be the colour classes in a proper colouring of . For , let denote an arbitrary linear ordering of . Let be the reverse of . Since we may assume that is large, Lemma 5 shows that there exists a set of -suitable permutations of for some .
For each , we introduce two 2-dimensional representations of . Let be the ordering of . Similarly, let be the ordering of . For each vertex in , say is the -th vertex in and is the -th vertex in . Then represent by the box with corners and . For each vertex , if has no neighbours in , then represent by the point ; otherwise, if is the leftmost neighbour of in and is the rightmost neighbour of in , then represent by the box with corners and , as illustrated in Figure 1. Now, in two new dimensions introduce the following representation. Represent each in by the box with corners and . For each vertex , if has no neighbours in , then represent by the point ; otherwise, if is the leftmost neighbour of in and is the rightmost neighbour of in , then represent by the box with corners and . In each of these four dimensions, add every vertex in with interval . Observe that and are both cliques in this representation.
By construction, for every edge of the boxes of and intersect in every dimension. Now consider a non-edge of with and . Let be the set of integers such that some neighbour of is in . Thus . Say is in . First suppose that . Since , for some permutation , we have for each . Let be the leftmost neighbour of in . Thus , and in the first 2-dimensional representation corresponding to , the right-hand-side of the box representing is to the left of the left-hand-side of the box representing , as illustrated in Figure 2(a). Thus the boxes representing and do not intersect. Now assume that . By construction, there is exactly one neighbour of in . Since , for some permutation , we have for each . If in , then , and as argued above and illustrated in Figure 2(b), the boxes representing and do not intersect. Otherwise, in . Then , and in the second 2-dimensional representation corresponding to , the right-hand-side of the box representing is to the left of the left-hand-side of the box representing . Hence the boxes representing and do not intersect. Therefore .
By Lemma 1,
Since and and ,
We now prove our first main result.
For every graph with maximum degree , as ,
Let and . By Corollary 9, there is a partition of , such that for each and . Note that
Applying Lemma 1 again,
For every poset whose comparability graph has maximum degree , as ,
Again, with (2), this gives
4. Euler Genus
We now prove our second main result.
For every graph with Euler genus , as .
By Lemma 4, contains a set of at most vertices such that . First suppose that . Deleting one vertex reduces boxicity by at most 1. Thus , and we are done since . Now assume that .
Let . Let be the set of vertices in with exactly one or exactly two neighbours in . Let . By Lemma 5, there is a 3-suitable set of permutations of for some . For each we introduce two dimensions, as illustrated in Figure 3. Represent each vertex by the box with corners and . For each vertex , if and are respectively the leftmost and rightmost neighbours of in , then represent by the box with corners and . Observe that and are both cliques in this representation. If and and , then the box representing intersects the box representing . Consider a non-edge with and . Since is 3-suitable and , for some , we have for each . Thus, for the 2-dimensional representation defined with respect to , the boxes representing and do not intersect.
Add each vertex in to every dimension with interval . We obtain a box representation of . Thus .
Let be the set of vertices in with at least three neighbours in . Let . Observe that .
To bound , we first bound , where . The number of edges in is least and at most by Euler’s formula. Thus , implying . Let . Let be an ordering of , where has minimum degree in . Define . Let be minimum such that has degree at least in . If is defined, then let and , otherwise let and .
Observe that .
Now consider . By construction, every vertex in has degree at most in . A permutation of catches a non-edge of with and if there are edges in , such that is between and in . Let . Let be random permutations of . For each non-edge of , the probability that catches equals . Thus, the probability that every catches is at most . Since the number of non-edges is at most , by the union bound, the probability that for some non-edge , every catches is at most . Hence, with positive probability, for every non-edge , some does not catch . Therefore, there exists permutations of , such that for every non-edge , some does not catch .
For each permutation we introduce two dimensions. Represent each vertex by the box with corners and . For each vertex , if has no neighbours in then represent by the point ; otherwise, if and are respectively the leftmost and rightmost neighbours of in , then represent by the box with corners and . Observe that and are both cliques in this representation. If and and , then the box representing intersects the box representing . For a non-edge with and , the box representing intersects the box representing if and only if catches . Since for every non-edge , some does not catch , the boxes representing and do not intersect. Thus .
5. Layered Treewidth
A tree decomposition of a graph is a set of non-empty sets (called bags) indexed by the nodes of a tree , such that for each vertex , the set induces a non-empty (connected) subtree of , and for each edge there is a node such that . The width of a tree decomposition is . The treewidth of a graph , denoted by , is the minimum width of a tree decomposition of . Treewidth is a key parameter in algorithmic and structural graph theory (see [37, 6, 26] for surveys). Chandran and Sivadasan  proved:
Theorem 15 ().
For every graph ,
A layering of a graph is a partition