of ordered pairs —calledconflicts— of colours from . The question is whether admits a colouring of the vertices so that no edge is in a conflict, i.e. there is no edge and conflict such that and . The authors in  and  also imposed further natural restrictions based on contrasting goals and perspectives, but here instead we only prescribe the maximum number of conflicts per edge.
In fact, this is equivalent to the “least conflict” version of the problem, with one conflict per edge, provided we pass to a multigraph of maximum edge multiplicity . Let us be more precise. Let be a multigraph. For any positive integer , a local -partition of is a collection of maps of the form , where denotes the set of edges incident to . So each is a partition of into parts, and for each the colour can be thought of as the local colour222By relabelling, we alternatively may define the as maps from to each image set of which contains at most elements, so not necessarily the same image for every . of associated to . Given such a collection , we say is conflict -colourable if there is some colouring of the vertices so that no edge has and . Observe we may assume each is onto, or else it is safe to colour with any colour not in the image. The (least) conflict choosability of is the smallest such that is conflict -colourable for any local -partition .
As we discuss in Section 2, conflict choosability considerably strengthens upon two notable list colouring parameters, separation choosability (cf. ) and adaptable choosability (cf. ), and so its study could potentially yield new insights into these two parameters.
Before continuing, we give two easy but instructive examples. First, for a square integer , consider two vertices with edges between them. Take the local -partition which lists all possible pairwise conflicts between the two vertices. So this is a -edge planar multigraph with maximum degree and multiplicity both that has conflict choosability strictly greater than . Second, for a positive integer , consider a star with centre and leaves , where each edge has multiplicity . Take the local -partition where the edges between and include all possible conflicts having as the local colour for . Since cannot be coloured, this is a -edge planar multigraph with maximum degree and maximum multiplicity that has conflict choosability strictly greater than .
Besides introducing conflict choosability and setting down some of its basic behaviour, our main task in this paper is to treat it in a classic setting for chromatic graph theory. We prove the following.
For some constant , if is a multigraph of maximum multiplicity that is embeddable on a surface of Euler genus , then .
The term cannot be improved by more than a constant factor due to the second example above, and we will see below that the other term is sharp up to at most a polylogarithmic factor.
Allow us to reiterate the case, which may be interpreted as an analogue of Heawood’s classic formula for the chromatic number .
There is a constant such that for every simple graph that is embeddable on a surface of Euler genus .
The case of Theorem 1 is of special interest, hinting at the following possible version of Heawood’s.
For any , there exists such that the following holds. For any simple graph that is embeddable on a surface of Euler genus , if every edge is assigned at most conflicts from , then is conflict -colourable, provided .
Theorem 1 follows from the following perhaps more general result.
For some constant , if is a multigraph with edges and maximum multiplicity , then .
If is a multigraph of maximum degree , then .
If is a multigraph of average degree , then .
The last two assertions alone highlight a clear distinction between conflict choosability and, say, ordinary choosability, for which the behaviour of the complete graphs is linear in , while that of the complete bipartite graphs is logarithmic in .
Notice that Proposition 6 helps to provide a broad certificate of sharpness of Theorems 1 and 4 up to polylogarithmic factors. This is akin to the two-vertex example exhibited earlier. In particular, consider the complete multigraph on vertices of uniform edge multiplicity . It is a -regular graph, so with edges, that has Euler genus. By Proposition 6, the conflict choosability is , and this is not far from the upper bound implied in both Theorems 1 and 4.
It may be challenging to eliminate the logarithmic factors in Theorems 1 and 4. Since we do not know the correct asymptotics in these results, we have made no effort to optimise the values of and . On the other hand, we managed to avoid the logarithmic factors for separation and adaptable choosability (see Theorems 11 and 12 below). The simpler argument uses Proposition 5 directly (rather than needing Lemma 13), and we present it in Section 4 as a warm up to proving our main result.
One might wonder if degeneracy could be an alternative way to prove Theorem 1, at least in the case. That was essentially Heawood’s original approach to bounding the chromatic number. As we will see in Section 2, density considerations have some use (see Lemma 7 below); however, a construction of Kostochka and Zhu  for the adaptable chromatic number shows that there are graphs of degeneracy which have conflict choosability greater than . There might yet be some constant such that the conflict choosability of any -degenerate graph on vertices is at most (which would imply the case of Theorem 1), but we have not been able to prove this thus far. Theorem 4 implies an upper bound of in this situation.
For small , it would be interesting to precisely determine the optimal upper bound on over all simple graphs embeddable on a surface of Euler genus . As we will indicate in Section 2 it is easy to verify that the extremal conflict choosability is for planar graphs.
1.1 Probabilistic preliminaries
We make use of the following basic probabilistic tools. We refer the reader to the monograph of Molloy and Reed  for further details.
The Chernoff Bound.
For any ,
The Lovász Local Lemma.
Consider a set of (bad) events such that for each
is mutually independent of a set of all but at most of the other events.
If , then with positive probability none of the events in
, then with positive probability none of the events inoccur.
The General Local Lemma.
Consider a set of (bad) events such that each is mutually independent of , for some . If we have reals such that for each
then the probability that none of the events in occur is at least .
In this section, we give some more definitions, one of conflict choosability, one of adaptable choosability, and one of separation choosability. We also show how these three parameters are related, and give a few comments related to planar graphs.
First we give an alternative definition of conflict choosability, which may be insightful. Let be a multigraph. Given a local -partition of , we say is conflict -orientable if there is some orientation of all edges of such that for every vertex , the set of local colours of associated to the (oriented) edges leaving does not contain all of . Then the conflict choosability of is the least such that is conflict -orientable for any local -partition .
Proof of equivalence.
Let and fix a local -partition of . It suffices to show that is conflict -orientable if and only if it is conflict -colourable. If it has a conflict -orientation, then for every choose a colour from that is absent from the local colours of associated to the edges leaving to produce a conflict -colouring. If it has a conflict -colouring , then orient towards all incident edges such that to produce a conflict -orientation. ∎
From this equivalence, the following proposition becomes plain.
If there is an orientation of such that every vertex has maximum outdegree less than , then .
This implies , cf. e.g. [1, Lem. 3.1].
If is a planar graph, then . If is a triangle-free planar graph, then . If is a a simple graph embeddable on a surface of Euler genus , then , where is Heawood’s formula for Euler genus .
Recall that every -degenerate graph has an orientation of maximum outdegree at most . So Proposition 7 cannot be improved in general, since there are -degenerate graphs with adaptable chromatic number greater than  (and, as we will shortly see, the same then is true of conflict choosability).
Next we discuss how conflict choosability is connected to two colouring parameters, both of which are weaker versions of list colouring, as introduced independently by Erdős, Rubin and Taylor  and by Vizing .
For completeness, we recall the classic definition. Let be a (multi)graph. For a positive integer , a mapping is called a -list-assignment of , and a colouring of is called an -colouring if for any . We say is -choosable if there is a proper -colouring of for any -list-assignment . The choosability of is the least such that is -choosable.
2.1 Adaptable choosability
The following list colouring parameter was proposed by Kostochka and Zhu . Let be a multigraph. Given a labelling of the edges, a (not-necessarily-proper) vertex colouring is adapted to if for every edge not all of , and are the same value. We say that is adaptably -choosable if for any -list-assignment and any labelling of the edges of , there is an -colouring of that is adapted to . The adaptable choosability of is the least such that is adaptably -choosable. Every proper colouring is adapted to any labelling , so always.
We observe adaptable choosability is at most conflict choosability.
For any multigraph , .
Fix and let . Let be a -list-assignment and let be a labelling of the edges of . For each , locally colour each edge incident to with colour if and . This yields a local -partition (as mentioned in the introduction, it is not important that the image of each map is equal to , the image of of each can be different sets of elements for each vertex ). By the choice of there must be a conflict -colouring. It follows from our definition of that this corresponds to an -colouring that is adapted to . ∎
We remark that adaptable choosability is in turn a strengthening of the adaptable chromatic number (for which the list assignment always takes all lists equal) and Hell and Zhu  have exhibited planar graphs with adaptable chromatic number at least . So conflict choosability is also exactly for such graphs.
2.2 Separation choosability
The following list colouring parameter was proposed by Kratochvíl, Tuza and Voigt . Let be a graph. We say a -list-assignment has maximum separation if for every edge of . We say is separation -choosable if there is a proper -colouring of for any -list-assignment that has maximum separation. The separation choosability of is the least such that is separation -choosable. Since the choosability of omits any separation requirement on the lists, always.
Let us see that separation choosability is at most adaptable choosability. This observation was made earlier , but we include it here for cohesion.
For any simple graph , .
Fix and let . Let be a -list-assignment of maximum separation. Let be a labelling defined for each by taking as the unique element of if it is nonempty, and arbitrary otherwise. By the choice of , there is guaranteed to be an -colouring that is adapted to . Due to the maximum separation property of and the definition of , the colouring must be proper. ∎
Conflict choosability is a direct strengthening of separation choosability, in the same way that “DP-colouring” is a strengthening of choosability .
Alternative proof that for any simple graph .
Fix and let . Let be a -list-assignment of maximum separation. Let be a local -partition of defined as follows. For each edge , if is the unique colour in , then let and . By the choice of , there is guaranteed to be a conflict -colouring . Due to the maximum separation property of and the definition of , the colouring is proper. ∎
We remark that Kratochvíl, Tuza and Voigt  proved that as by the use of affine planes. This is enough to certify sharpness of our Theorems 1 and 4 each up to a logarithmic factor (and Proposition 5 up to a constant factor) for simple graphs.
We also note that Škrekovski  conjectured that every planar graph has separation choosability at most , but this remains open to the best of our knowledge. If true, it would imply that separation choosability and adaptable choosability can be distinct for some planar graphs.
Proof of Proposition 5.
Let be a multigraph of maximum degree and fix . Let be a local -partition of . Consider a random colouring where each vertex is given an independent uniform choice. For each edge , let be the event that and . For all , and is mutually independent of all but at most other events . Observe that is a conflict -colouring if and only if all the events do not occur. The Lovász Local Lemma guarantees with positive probability a conflict -colouring if , which follows from the choice of . ∎
Note that the bound in Proposition 5 can be slightly improved to using the Local Cut Lemma [3, Theorem 3.1] instead of the Lovász Local Lemma, using the same set of bad events. We have deliberately chosen to present the simpler, weaker bound.
The following proof is analogous to one in .
Proof of Proposition 6.
Let be a multigraph of average degree , where and . Let and consider a random local -partition of where, for each edge , the pair is independently, uniformly chosen from pairs in . For any fixed , is a conflict -colouring with probability . By the union bound and Markov’s inequality, the probability that is conflict -colourable is at most . Since has average degree , we have by the choice of that . This implies and so . We have thus shown that with positive probability there is a local -partition for which is not conflict -colourable. ∎
4 Proof of Theorem 1
As a warm up to the main proof, we show the following result, an adaptable choosability analogue of Theorem 4.
If is a multigraph with edges and maximum multiplicity , then .
The proof of Theorem 11 can be viewed as a simplified version of the proof of Theorem 4. Afterwards, we show how the following result, an adaptable choosability analogue of Theorem 1, is a consequence of Theorem 11. (At the same time, we also show how Theorem 4 implies Theorem 1.)
For some constant , if is a multigraph of maximum multiplicity that is embeddable on a surface of Euler genus , then .
Theorems 11 and 12 imply the same bounds for separation choosability, and both are sharp up to the choice of due to the complete graphs with uniform edge multiplicity . Let us mention that the question of whether graphs of Euler genus have adaptable chromatic and choice numbers at most of order was first raised in December of 2007 during the Graph Theory 2007 meeting in Fredericia, Denmark.
Proof of Theorem 11.
Let be a multigraph with and maximum multiplicity . Let , let be a -list-assignment, and consider any labelling of the edges of . We want to prove that there is an -colouring of that is adapted to . We can assume that is connected (or else we consider each component separately), and in particular has vertices.
Let , and let be chosen uniformly at random. Set . For any and ,
is binomially distributed with parameter. The Chernoff Bound implies that with probability at most , where the first inequality uses . By a union bound, there is a bipartition such that for any and .
Let be the set of vertices of degree at least in and let . Since , has most vertices, and thus has maximum degree at most . By definition, also has maximum degree at most . We remove all the colours of from for each , and all the colours of from for each . After this operation, each list has at least colours left. Since , it follows from Proposition 5 that has an -colouring adapted to using only colours from while has an -colouring adapted to using only colours from . Since and are disjoint, we obtain an -colouring of adapted to , as desired. ∎
Assume for a contradiction that there is a counterexample to Theorem 1 or 12. Take in such way that is minimised, and subject to this the number of vertices of is minimised. We can assume that is connected (or else we consider each component separately). Let be the simple graph underlying . By the minimality of , has no embedding on a surface of smaller Euler genus, and thus has a cellular embedding on a surface of Euler genus . It follows from Euler’s Formula that has edges, and so has edges. Let (for Theorem 1) or (for Theorem 12), and assume that each vertex has local colours. If has a vertex of degree less than , then remove . By the minimality of , we can colour and then find a suitable colour for (since has at least local colours and fewer than neighbours in ). Thus, we can assume that has minimum degree at least , and thus at least edges. Consequently, .
To prove Theorem 4, we require the following slightly technical result.
For any , let be a multigraph with a vertex partition such that
the induced submultigraph has maximum degree at most ,
all vertices in have maximum degree at most in , and
all vertices in have maximum degree at most in .
There is a constant such that for any local -partition of , where , there is a colouring such that is a conflict -colouring of and no vertex has more than incident edges , , such that .
Proof of Theorem 4.
Let be a multigraph with edges and maximum multiplicity . Let be the set of vertices of degree at least in and let . Since , has at most vertices, and thus has maximum degree at most . It follows from the definition of that also has maximum degree at most . Note that is trivially at most . So by a large enough fixed choice of we may assume is large enough so that the conditions of Lemma 13 are satisfied with . Let be the constant associated to the corresponding application of Lemma 13. Let be an integer at least and let be a local -partition of . It follows from an application of Lemma 13 that there is a conflict -colouring of . It remains to colour in such a way that it is compatible with .
For each vertex , remove from any edge incident to if there exists some incident edge , , such that and . We also (locally) remove each of the colours associated to the edges we removed. By one of the properties of guaranteed by Lemma 13, this process removes at most of the colours incident to each vertex in . By arbitrarily deleting any excess local colours as well as any of the incident edges with those colours, then relabelling colours, we are left with a local -partition of a submultigraph of with maximum degree at most , where . By Proposition 5, this submultigraph admits a conflict -colouring . The colour and edge removal process we performed ensures that, by reversing the relabelling, corresponds to a conflict -colouring of that combines with to produce a conflict -colouring of all of . ∎
Proof of Lemma 13.
Let where is some constant large enough to guarantee certain properties as specified later in the proof. Let be a local -partition of .
We must do a pruning operation before proceeding —in fact, this is the crucial step in the proof. By taking large enough, we may assume for each and each that the number of edges in with its other endpoint also in is at most . (We summarily remove all edges associated to every colour not satisfying the property, and since the maximum degree of is at most this removes at most of the colours around each vertex in .)
Let . Consider a random selection of colours where each of the local colours is selected according to an independent Bernoulli trial of probability . With an eye to applying the General Local Lemma, let us define three types of (bad) events.
For a vertex , none of the colours around is selected.
For an edge with , and are both selected.
For a vertex , there are more than edges , , for which is selected.
If we obtain a selection for which none of the above events occurs, then we are done. This is because the deselection of a colour does not introduce any new event of Type II or III. So we can arbitrarily deselect all but one of the colours around each vertex, and the remaining selection induces the desired colouring , thanks to the fact that no events of Type II or III hold.
For each , the probability of a Type I event is if is chosen large enough. For each edge , , the probability of a Type II event is . For each vertex , the probability of a Type III event is at most by the Chernoff Bound.
The choice to generate the random colouring according to independent Bernoulli trials rather than a uniform colour per vertex (as in Proposition 5) is important for us in establishing the following bounds on dependence between bad events, especially for Type III events. Each Type I event is mutually independent of all but at most events of Type I, at most events of Type II, and at most events of Type III. Each Type II event is mutually independent of all but at most events of Type I, at most events of Type II, and at most events of Type III. By the pruning operation we did at the beginning, each Type III event is mutually independent of all but at most events of Type I, at most events of Type II, and at most events of Type III. (To be more explicit, each Type III event is determined by up to
independent Bernoulli random variables, each of which corresponds to a local colour of a neighbour. Thanks to the pruning, the number of Type II events, say, that also use this randomness is at most. The Type III event is mutually independent of all other Type II events.)
We associate weight to each event of Type I or II, and weight to each event of Type III. By the considerations above, the General Local Lemma guarantees the desired selection of colours with positive probability, provided the following three inequalities hold (where we repeatedly used that if ):
It is straightforward to check that suffices. ∎
The above proof can be straightforwardly adapted for the same upper bound (with a larger constant ) on a stronger type of conflict choosability where additionally we must assign distinct colours per vertex. What this then directly implies is that, for any simple graph that is embeddable on a surface of Euler genus , the conflict choosability is even if we allow conflicts per edge and demand distinct colours per vertex.
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