Packing graphs of bounded codegree
PPacking graphs of bounded codegree
Wouter Cames van Batenburg ∗ Ross J. Kang ∗ September 30, 2018
Abstract
Two graphs G and G on n vertices are said to pack if there exist injective mappings oftheir vertex sets into [ n ] such that the images of their edge sets are disjoint. A longstandingconjecture due to Bollob´as and Eldridge and, independently, Catlin, asserts that, if (∆ ( G ) +1)(∆ ( G ) + 1) ≤ n + 1, then G and G pack. We consider the validity of this assertion underthe additional assumption that G or G has bounded codegree. In particular, we prove forall t ≥ G contains no copy of the complete bipartite graph K ,t and ∆ > t · ∆ ,then (∆ ( G ) + 1)(∆ ( G ) + 1) ≤ n + 1 implies that G and G pack. We also provide a mildimprovement if moreover G contains no copy of the complete tripartite graph K , ,s , s ≥ Let G and G be graphs on n vertices. (All graphs are assumed to have neither loops nor multipleedges.) We say that G and G pack if there exist injective mappings of their vertex sets into[ n ] = { , . . . , n } so that their edge sets have disjoint images. Equivalently, G and G pack if G isa subgraph of the complement of G . The maximum codegree ∆ ∧ ( G ) of a graph G is the maximumover all vertex pairs of their common degree, i.e. ∆ ∧ ( G ) < t if and only if G contains no copy ofthe complete bipartite graph K ,t . The maximum adjacent codegree ∆ (cid:77) ( G ) of G is the maximumover all pairs of adjacent vertices of their common degree, i.e. ∆ (cid:77) ( G ) < s if and only if G containsno copy of the complete tripartite graph K , ,s . Clearly, ∆ (cid:77) ( G ) ≤ ∆ ∧ ( G ) always. We let ∆ and∆ denote the maximum degrees of G and G , respectively, and ∆ ∧ and ∆ (cid:77) the correspondingmaximum (adjacent) codegrees. We provide sufficient conditions for G and G to pack in terms of∆ , ∆ , ∆ ∧ , ∆ (cid:77) .For integers t ≥ ≥
1, we define α ∗ ( t, ∆ ) := 12 (2 + γ + (cid:112) γ + γ ) , where γ = ∆ ∆ + 1 · t − t . Note α ∗ is the larger solution to the equation ( α − − γα = 0 and (9 + √ ≤ α ∗ ≤ (3 + √ Theorem 1.1
Let G and G be graphs on n vertices with respective maximum degrees ∆ and ∆ . Let ∆ ∧ bethe maximum codegree of G . Let t ≥ be an integer and let α > α ∗ = α ∗ ( t, ∆ ) and < (cid:15) < / be reals. Then G and G pack if ∆ ∧ < t and n is larger than each of the following quantities: (cid:18) t + α ( α − α − − α (cid:19) · ∆ + ∆ ∆ , (1)(2 αt + 2) · ∆ + ((2 α + 1) t − · ∆ + (1 − (cid:15) ) · ∆ ∆ , (2)1 + (cid:18) (cid:15) − (cid:15) (cid:19) · ∆ + ∆ ∆ , and (3) (cid:18) t + 3 − (cid:15) (cid:19) · ∆ + 3 − (cid:15) t − · ∆ + 1 + (cid:15) · ∆ ∆ . (4) ∗ Department of Mathematics, Radboud University Nijmegen, Postbus 9010, 6500 GL Nijmegen, The Netherlands. [email protected] , [email protected] . a r X i v : . [ m a t h . C O ] M a y heorem 1.2 Let G and G be graphs on n vertices with respective maximum degrees ∆ and ∆ . Let ∆ ∧ bethe maximum codegree of G and ∆ (cid:77) the maximum adjacent codegree of G . Let s ≥ and t ≥ be integers and let α > α ∗ = α ∗ ( t, ∆ ) be real. Then G and G pack if ∆ ∧ < t , ∆ (cid:77) < s , and n islarger than both of the following quantities: (cid:18) t + α ( α − α − − α (cid:19) · ∆ + ∆ ∆ and (5)(2 + 2 αt ) · ∆ + ( s − · ∆ + ((2 α + 1) t − · ∆ . (6)For better context, we compare Theorems 1.1 and 1.2 to a line of work on graph packing that wasinitiated in the 1970s [2, 5, 6, 15]. The following is a central problem in the area. Conjecture 1.3 (Bollob´as and Eldridge [2] and Catlin [6])
Let G and G be graphs on n vertices with respective maximum degrees ∆ and ∆ . Then G and G pack if (∆ + 1)(∆ + 1) ≤ n + 1 . If true, the statement would be sharp and would significantly generalise a celebrated result of Hajnaland Szemer´edi [10] on equitable colourings. Sauer and Spencer [15] showed that 2∆ ∆ < n is asufficient condition for G and G to pack, which is seen to be sharp when one of the graphs isa perfect matching. Thus far the Bollob´as–Eldridge–Catlin (BEC) conjecture has been confirmedin the following special cases: ∆ = 2 [1]; ∆ = 3 and n sufficiently large [9]; G bipartite and n sufficiently large [8]; and G d -degenerate, ∆ ≥ d and ∆ ≥
215 [4]. Moreover, an approximateBEC condition, (∆ + 1)(∆ + 1) ≤ n/ G and G to pack, provided that∆ , ∆ ≥
300 [13]. Theorem 1.1 implies the following.
Corollary 1.4
Let G , G , ∆ , ∆ and ∆ ∧ be as before. Let t ≥ be an integer. Then G and G pack if ∆ ∆ + ∆ ≤ n + 1 and ∆ ∧ < t and ∆ > t · ∆ . Proof.
Choose (cid:15) = (2 t − / (4 t −
3) and α = 3 in Theorem 1.1. Using that ∆ > t ∆ > (4 t − t − t − · ∆ , it follows that max((1) , (2) , (3) , (4)) ≤ (∆ + 1)(∆ + 1) − ≤ n . So G and G pack.The following results concerning the BEC-conjecture follow immediately. Corollary 1.5
Given an integer t ≥ , the BEC conjecture holds under the additional condition that the maximumcodegree ∆ ∧ of G is less than t and ∆ > t · ∆ . We were unable to avoid the linear dependence on ∆ in the lower bound condition on ∆ . Althoughwe have not seriously attempted to optimise the factor 17 t above, Theorem 1.2 improves on thisfactor under the additional assumption that ∆ (cid:77) is bounded, as exemplified by the following corollary. Corollary 1.6
Given an integer t ≥ , the BEC conjecture holds under the additional condition that the maximumcodegree ∆ ∧ of G is less than t , G is triangle-free, and ∆ > (4 + √ t · ∆ . Proof.
Choose α = t (6 t + 1 + √ t + 4 t + 1) and s = 1 in Theorem 1.2. Using that t + α ( α − α − − α − α + 1) t − > (4 + √ t · ∆ > ((2 α + 1) t − · ∆ , it follows that max((5) , (6)) ≤ (∆ + 1)(∆ + 1) − ≤ n . So G and G pack. 2 tructure of the paper In the next section, we provide some notation and preliminary observations. In Section 3, we discussthe common features of a hypothetical critical counterexample to one of our theorems. In Section 4,we prove Theorems 1.1 and 1.2. We conclude the paper with some remarks about the results, proofsand further possibilities.
Here we introduce some terminology which we use throughout. We often call G the blue graphand G the red graph. We treat the injective vertex mappings as labellings of the vertices from1 to n . However, rather than saying, “the vertex in G (or G ) corresponding to the label i ”,we often only say, “vertex i ”, since this should never cause any confusion. Our proofs rely onaccurately specifying the neighbourhood structure as viewed from a particular vertex. Let i ∈ [ n ].The blue neighbourhood N ( i ) of i is the set { j | ij ∈ E ( G ) } and the blue degree deg ( i ) of i is | N ( i ) | . The red neighbourhood N ( i ) and red degree N ( i ) are defined analogously. For j ∈ [ n ], a red–blue-link (or – -link) from i to j is a vertex i (cid:48) such that ii (cid:48) ∈ E ( G ) and i (cid:48) j ∈ E ( G ). The red–blue-neighbourhood N ( N ( i )) of i is the set { j | ∃ red–blue-link from i to j } . A blue–red-link(or – -link) and the blue–red-neighbourhood N ( N ( i )) are defined analogously.In search of a certificate that G and G pack, without loss of generality, we keep the vertex labellingof the blue graph G fixed, and permute only the labels in the red graph G . This can be thought ofas “moving” the red graph above a fixed ground set [ n ]. In particular, we seek to avoid the situationthat there are i, j ∈ [ n ] for which ij is an edge in both G and G — in this situation, we call ij a purple edge induced by the labellings of G and G . So G and G pack if and only if theyadmit a pair of vertex labellings that induces no purple edge. In our search, we make small cyclicsub-permutations of the labels (of G ), which are referred to as follows. For i , . . . , i (cid:96) − ∈ [ n ], a( i , . . . , i (cid:96) − ) -swap is a relabelling of G so that for each k ∈ { , . . . , (cid:96) − } the vertex labelled i k isre-assigned the label i k +1 mod (cid:96) . In fact, we shall only require swaps having (cid:96) ∈ { , } . The followingobservation describes when a swap could be helpful in the search for a packing certificate. This isidentical to Lemma 1 in [13]. Lemma 2.1
Let u , . . . , u (cid:96) − ∈ [ n ] . For every k, k (cid:48) ∈ { , . . . , (cid:96) − } , suppose that there is no red–blue-link from u k to u k +1 mod (cid:96) and that, if u k u k (cid:48) ∈ E ( G ) , then u ( k +1 mod (cid:96) ) u ( k (cid:48) +1 mod (cid:96) ) / ∈ E ( G ) . Then there isno purple edge incident to any of u , . . . , u (cid:96) − after a ( u , . . . , u (cid:96) − ) -swap. We will use a classic extremal set theoretic result to upper bound the size of certain vertex subsets.
Lemma 2.2 (Corr´adi [7])
Let A , . . . , A N be k -element sets and X be their union. If | A i ∩ A j | ≤ t − for all i (cid:54) = j , then | X | ≥ k N/ ( k + ( N − t − . In particular, this implies the following.
Corollary 2.3
Let A , . . . , A N be size ≥ k subsets of a set X . If k > ( t − · | X | and | A i ∩ A j | ≤ t − for all i (cid:54) = j , then N ≤ | X | · k − ( t − k − ( t − · | X | . Proof.
Consider arbitrary subsets A ∗ ⊂ A , . . . , A ∗ N ⊂ A N of size k . An application of Corr´adi’slemma to A ∗ , . . . , A ∗ N yields that | X | ≥ k · N/ ( k + ( N − t − k − ( t − · | X | ) · N ≤ ( k − t ) · | X | . The corollary follows after dividing both sides3 u Figure 1: All vertices (except possibly v ) are reachable by a link from u (Claim 3.1).of the inequality by k − ( t − · | X | . Note that this division does not cause a sign change becauseof the assumption that k > ( t − · | X | . The overall proof structure we use for both theorems is the same, and in this section we describecommon features and some further notation. Suppose the theorem (one of Theorem 1.1 or 1.2) isfalse. Then there must exist a counterexample, that is, a pair ( G , G ) of non-packable graphs on n vertices that satisfy the conditions of the theorem.Moreover, we may assume that ( G , G ) is a critical pair in the sense that G is edge-minimal amongall counterexamples. In other words, G and G − e pack for any e ∈ E ( G ). There is no loss ofgenerality, since the removal of an edge from G increases neither ∆ nor ∆ (cid:77) and obviously affectsnone of ∆ , ∆ ∧ and n , thus maintaining the required conditions.Now choose any edge e = uv ∈ E ( G ). Criticality implies that there is a pair of labellings of G and G such that e is the unique purple edge, for otherwise G and G − e do not pack. Let us fixsuch a pair of labellings so that we can further describe the neighbourhood structure as viewed from u (or v ). Estimation of the sizes of subsets in this neighbourhood structure is our main method forderiving upper bounds on n that in turn yield the desired contradiction from which the theoremfollows.We need the definition of the following vertex subsets (which are analogously defined for v also): A ( u ) := N ( N ( u )) \ ( N ( u ) ∪ N ( u ) ∪ N ( N ( u ))) ,B ( u ) := N ( N ( u )) \ ( N ( u ) ∪ N ( u ) ∪ N ( N ( u ))) ,A ∗ ( u ) := N ( N ( u )) \ ( N ( u ) ∪ N ( N ( u ))) , and N ∗ ( u ) := N ( u ) ∩ ( N ( N ( u )) \ ( N ( u ) ∪ N ( N ( u )))) . One justification for specifying the above subsets is that the following two claims (which are essen-tially Claims 1 and 2 in [13]) hold.
Claim 3.1
For all w ∈ [ n ] \ { v } , there is a red–blue-link or a blue–red-link from u to w . Proof.
If not, then by Lemma 2.1, a ( u, w )-swap yields a new labelling such that uv is not purpleanymore and no new purple edges are created. Thus G and G pack, a contradiction. See Figure 1.4 = N ( u ) ∩ N ( u ) uN ( u ) N ( u ) B ( u ) A ( u )Figure 2: The neighbourhood structure of a hypothetical critical counterexample, as seen from u . Claim 3.2
For all a ∈ A ∗ ( u ) and b ∈ B ( u ) , there is a red–blue-link from a to b . Proof.
Since B ( u ) ∩ N ( u ) = B ( u ) ∩ N ( u ) = ∅ and A ∗ ( u ) ∩ N ( u ) = ∅ , we have that bu / ∈ E ( G ) ∪ E ( G ) and ua / ∈ E ( G ). Furthermore, since A ∗ ( u ) ∩ N ( N ( u )) = B ( u ) ∩ N ( N ( u )) = ∅ ,there is no red–blue-link from u to a or from b to u . Now suppose that there is also no red–blue-linkfrom a to b . Then it follows from Lemma 2.1 that after a ( u, a, b )-swap there is no purple edgeincident to any of u, a, b , which implies that there is no purple edge at all. So we have obtained apacking of G and G , a contradiction.In the next claim, we list three upper bounds on the total number n of vertices in terms of the sizesof the vertex subsets defined above. In the proofs of Theorems 1.1 and 1.2, we consider several casesfor which we prove at least one of these upper bounds to be small enough for a contradiction withthe assumed lower bounds on n . Claim 3.3
The total number n of vertices is at most each of the following quantities: ( i ) | N ( u ) | + | A ∗ ( u ) | + | N ( N ( u )) | , ( ii ) | N ∗ ( u ) | + | N ( u ) | + | B ( u ) | + | N ( N ( u )) | , ( iii ) | A ∗ ( v ) | + | A ∗ ( u ) | + | ( N ( u ) ∪ N ( N ( u ))) ∩ ( N ( v ) ∪ N ( N ( v ))) | . Proof.
In all cases, [ n ] equals the union of the neighbourhood sets that occur in the upper bound.( i ) The union of N ( u ), A ∗ ( u ) and N ( N ( u )) covers { v } ∪ N ( N ( u )) ∪ N ( N ( u )), which byClaim 3.1 equals [ n ].( ii ) The union of N ∗ ( u ), N ( u ), B ( u ) and N ( N ( u )) covers { v } ∪ N ( N ( u )) ∪ N ( N ( u )), whichequals [ n ].( iii ) By the proof of (i), [ n ] is the union of A ∗ ( u ) and N ( u ) ∪ N ( N ( u )) as well as the unionof A ∗ ( v ) and N ( v ) ∪ N ( N ( v )). It follows that [ n ] also is the union of A ∗ ( u ), A ∗ ( v ) and( N ( u ) ∪ N ( N ( u ))) ∩ ( N ( v ) ∪ N ( N ( v ))).The reason for working with N ∗ ( u ) and A ∗ ( u ) rather than the simpler sets N ( u ) and A ( u ) is thefollowing. Under the requirement that the codegree ∆ ∧ of G is less than t , we can upper bound | N ∗ ( u ) | entirely in terms of ∆ . This is sharper than the trivial bound | N ( u ) | ≤ ∆ because wework under conditions with ∆ rather larger than ∆ . Similarly, since N ∗ ( u ) ⊂ N ( u ), we need tocompensate for the loss of covered vertices by working with the slightly enlarged set A ∗ ( u ), ratherthan A ( u ). The following claims use the condition ∆ ∧ < t (which is assumed by both theorems).5 laim 3.4 | N ∗ ( u ) | ≤ ( t − · ∆ . Proof.
Suppose | N ( u ) ∩ N ( N ( u )) | ≥ ( t − · ∆ + 1, then there is at least one x ∈ N ( u ) suchthat | N ( u ) ∩ N ( x ) | ≥ | N ( u ) | · (( t − · ∆ + 1) > t −
1, which contradicts ∆ ∧ < t .The following claim (in combination with Corr´adi’s lemma) is useful for an upper bound on | B ( u ) | that is only linear in ∆ , provided that | A ∗ ( u ) | is at least quadratic in ∆ . See Case ( i ) in the proofof Theorem 1.1. Claim 3.5
For any b ∈ B ( u ) , | N ( b ) ∩ A ∗ ( u ) | ≥ | A ∗ ( u ) | / ∆ − t (∆ + 1) . Proof.
For all b ∈ N ( N ( u )) it holds that | N ( b ) ∩ N ( N ( u )) | ≤ ( t − · | N ( u ) | ≤ ( t − · ∆ .Indeed, otherwise there would exist a blue copy of K ,t in the graph induced by N ( N ( u )) ∪ N ( u ).Similarly, | N ( b ) ∩ N ( u ) | ≤ t and | N ( b ) ∩ N ( u ) | ≤ ∆ . So for every b ∈ N ( N ( u )), at most t · (∆ + 1) blue neighbours of b are in [ n ] \ A ( u ). So in particular, for every b ∈ B ( u ), at most t · (∆ + 1) blue neighbours of b are in [ n ] \ A ∗ ( u ).Using Claim 3.2 and the fact that each blue neighbour of a fixed b ∈ B ( u ) has at most ∆ redneighbours in A ∗ ( u ), we see that every b ∈ B ( u ) has at least (cid:100)| A ∗ ( u ) | / ∆ (cid:101) blue neighbours, andthus at least | A ∗ ( u ) | / ∆ − t (∆ + 1) blue neighbours in A ∗ ( u ). Suppose the theorem is false. Consider a critical counterexample, a pair of non-packable graphs( G , G ), with G edge-minimal, satisfying the constraints of the theorem. We distinguish threecases, for each of which we derive an upper bound on n , given by one of the inequalities (8), (10)and (16). At least one of these three inequalities should hold, so together they contradict thecondition that max ((8) , (10) , (16)) = max((1) , (2) , (3) , (4)) < n , thus proving the theorem.( i ) There exists a vertex u ∈ [ n ] and there are labellings of G and G such that u is incident tothe unique purple edge and | A ∗ ( u ) | ≥ αt · ∆ (∆ + 1).( ii ) Case ( i ) does not hold and furthermore | N ( u ) ∩ N ( v ) | < (1 − (cid:15) ) · ∆ for some edge uv ∈ E ( G ).( iii ) Neither of Cases ( i ) and ( ii ) hold.We now proceed with deriving upper bounds on n for each of these three cases. Bound for Case ( i ) . Choose a vertex u ∈ [ n ] and labellings of G and G such that u is incidentto the unique purple edge and | A ∗ ( u ) | ≥ αt · ∆ (∆ + 1). See Figure 3 for a depiction of theargumentation in this case. From now on, we write k := | A ∗ ( u ) | / ∆ − t (∆ + 1). Our first tool isClaim 3.5, which yields that all b ∈ B ( u ) satisfy | N ( b ) ∩ A ∗ ( u ) | ≥ k . Note that k ≥
1, since α > X = A ∗ ( u ) and N = | B ( u ) | and with size ≥ k subsets A , . . . , A N ⊂ X given by N ( b ) ∩ A ∗ ( u ), for all b ∈ B ( u ).Note that | A i ∩ A j | ≤ t − i (cid:54) = j , or else there would be a blue copy of K ,t .In order to apply Corollary 2.3, we need to check that its condition k > ( t − · | A ∗ ( u ) | holds. Forthat, we write β := | A ∗ ( u ) | / ( t ∆ (∆ + 1)), so that k = ( β − t (∆ + 1). Now k − ( t − · | A ∗ ( u ) | = (( β − t (∆ + 1)) − βt ∆ (∆ + 1)( t − (cid:0) ( β − − γ · β (cid:1) · ( t (∆ + 1)) , uB ( u ) verysmall A ∗ ( u ) large Figure 3: A depiction of Case ( i ) of Theorem 1.1, that | A ∗ ( u ) | = Ω(∆ ) implies | B ( u ) | = O (∆ ).which is positive if and only if ( β − − γβ >
0, which holds true because β ≥ α > α ∗ . Thus, byCorollary 2.3, we obtain | B ( u ) | ≤ | A ∗ ( u ) | · k − ( t − k − ( t − · | A ∗ ( u ) | = 1 − t − kk | A ∗ ( u ) | − t − k . The numerator and denominator of the right hand side are both positive, so we can bound andrearrange as follows: | B ( u ) | ≤ (cid:18) k | A ∗ ( u ) | − t − k (cid:19) − = (cid:18) ( β − t (∆ + 1) βt ∆ (∆ + 1) − t − β − t (∆ + 1) (cid:19) − = ∆ · (cid:18) β − β − β − · ∆ ∆ + 1 · t − t (cid:19) − = ∆ · (cid:18) β − β − γβ − (cid:19) − ≤ ∆ · α ( α − α − − γα , (7)where the last step holds because β ≥ α > α ∗ and α ∗ is the larger singular point of β ( β − β − − γβ ,which is a decreasing function of β for all β > α ∗ .Evaluating (7) and Claim 3.4 in the upper bound of Claim 3.3( ii ) yields n ≤ | N ∗ ( u ) | + | N ( u ) | + | B ( u ) | + | N ( N ( u )) |≤ ( t − · ∆ + ∆ + α ( α − α − − α · ∆ + ∆ ∆ = (cid:18) t + α ( α − α − − α (cid:19) · ∆ + ∆ ∆ . (8) Bound for Case ( ii ) . Choose labellings of G and G such that there is a unique purple edge uv that satisfies | N ( u ) ∩ N ( v ) | < (1 − (cid:15) ) · ∆ . Note that the inequalities | A ∗ ( u ) | < αt · ∆ (∆ + 1)and | A ∗ ( v ) | < αt · ∆ (∆ + 1) are satisfied as well, as a direct consequence of the assumptions ofCase ( ii ).We proceed with deriving a technical estimate on an intersection of neighbourhood sets. For each x ∈ N ( u ) \ N ( v ) and y ∈ N ( v ) \ N ( u ) we have x (cid:54) = y and therefore absence of blue copies of K ,t uA ∗ ( v ) small A ∗ ( u ) small N ( N ( u ) ∩ N ( v )) small N ( N ( u ) \ N ( v )) ∩ N ( N ( v ) \ N ( u )) small Figure 4: A depiction of Case ( ii ) of Theorem 1.1, that | N ( N ( u )) ∩ N ( N ( v )) | is small.implies the inequality | N ( x ) ∩ N ( y ) | ≤ t −
1. So | N ( N ( u ) \ N ( v )) ∩ N ( N ( v ) \ N ( u )) | ≤ (cid:88) x ∈ N ( u ) \ N ( v ) (cid:88) y ∈ N ( v ) \ N ( u ) | N ( x ) ∩ N ( y ) |≤ | N ( u ) \ N ( v ) | · | N ( v ) \ N ( u ) | · ( t − ≤ (∆ − | N ( u ) ∩ N ( v ) | ) · ( t − . Furthermore, since | N ( u ) ∩ N ( v ) | < (1 − (cid:15) ) · ∆ , | N ( N ( u )) ∩ N ( N ( v )) | ≤ | N ( N ( u ) ∩ N ( v )) | + | N ( N ( u ) \ N ( v )) ∩ N ( N ( v ) \ N ( u )) | < ∆ · | N ( u ) ∩ N ( v ) | + (∆ − | N ( u ) ∩ N ( v ) | ) · ( t − ≤ max p ∈{ , , ,..., (cid:98) (1 − (cid:15) ) · ∆ (cid:99)} (cid:0) ∆ · p + (∆ − p ) · ( t − (cid:1) . See Figure 4. Finally, we evaluate this in Claim 3.3( iii ) to find the following bound on n : n ≤ | A ∗ ( v ) | + | A ∗ ( u ) | + | ( N ( u ) ∪ N ( N ( u ))) ∩ ( N ( v ) ∪ N ( N ( v ))) |≤ | A ∗ ( v ) | + | A ∗ ( u ) | + | N ( u ) | + | N ( v ) | + | N ( N ( u )) ∩ N ( N ( v )) |≤ αt · ∆ (∆ + 1) + 2∆ + max p ∈{ , , ,..., (cid:98) (1 − (cid:15) ) · ∆ (cid:99)} (cid:0) ∆ · p + (∆ − p ) · ( t − (cid:1) . (9)In particular, this implies the slightly rougher bound n ≤ αt · ∆ (∆ + 1) + 2∆ + (1 − (cid:15) ) · ∆ ∆ + ∆ · ( t − . (10) Bound for Case ( iii ) . Choose a pair of labellings of G and G that induces a unique purple edge uv . The assumptions of this case imply, in particular, that in the red graph the neighbourhoods ofeach pair of adjacent vertices overlap significantly: | N ( x ) ∩ N ( y ) | ≥ (1 − (cid:15) ) · ∆ for each xy ∈ E ( G ).We will derive two consequences, namely the implication (cid:18) | A ∗ ( u ) | ≥ + (cid:15) · ∆ − (cid:15) (cid:19) = ⇒ (cid:0) | B ( u ) | ≤ ( t − · ∆ (cid:1) (11)8 u aB ( u ) small A ∗ ( u ) not very small Figure 5: A depiction of (11) in Case ( iii ) of Theorem 1.1.and the inequality | N ( N ( u )) | ≤ (cid:15) ∆ + 1 − (cid:15) t − · ∆ + 32 ∆ . (12)We start with proving the statement (11), the first consequence. See Figure 5. Suppose a ∈ A ∗ ( u ) \ N ( u ) has a red neighbour x ∈ N ( u ). Then ux and ax are edges of G , so | N ( a ) ∩ N ( x ) | ≥ (1 − (cid:15) )∆ and | N ( u ) ∩ N ( x ) | ≥ (1 − (cid:15) )∆ . Combining this with the obvious fact that | N ( x ) | ≤ ∆ yieldsthat | N ( a ) ∩ N ( u ) | ≥ (1 − (cid:15) ) · ∆ . (13)Let us define A ∗∗ ( u ) := { a ∈ A ∗ ( u ) \ N ( u ) | a has a red neighbour in N ( u ) } . It follows from (13) that (cid:80) a ∈ A ∗∗ ( u ) | N ( a ) ∩ N ( u ) | ≥ | A ∗∗ ( u ) | · (1 − (cid:15) ) · ∆ , so (cid:88) x ∈ N ( u ) | N ( x ) | ≥ (cid:88) x ∈ N ( u ) | N ( x ) ∩ N ( u ) | + (cid:88) a ∈ A ∗∗ ( u ) | N ( a ) ∩ N ( u ) |≥ (1 − (cid:15) )∆ · | N ( u ) | + | A ∗∗ ( u ) | · (1 − (cid:15) ) · ∆ , and (crucially) since (cid:80) x ∈ N ( u ) | N ( x ) | ≤ ∆ · | N ( u ) | , it follows that | A ∗∗ ( u ) | ≤ | N ( u ) | · ∆ − (1 − (cid:15) ) · ∆ | N ( u ) | (1 − (cid:15) ) · ∆ = (cid:15) · | N ( u ) | − (cid:15) . (14)Next, suppose we would have that | A ∗ ( u ) | ≥ | N ( u ) | + | A ∗∗ ( u ) | . Then there exists a vertex a ∈ A ∗ ( u ) \ ( N ( u ) ∪ A ∗∗ ( u )). By the definition of A ∗∗ ( u ), this vertex satisfies N ( a ) ∩ N ( u ) = ∅ .Furthermore, since a ∈ A ∗ ( u ), we have that for all b ∈ B ( u ) there is a red–blue-link from a to b . In other words, B ( u ) = N ( N ( a )) ∩ B ( u ). This implies that | B ( u ) | = | N ( N ( a )) ∩ B ( u ) | ≤| N ( N ( a )) ∩ N ( N ( u )) | ≤ ( t − · ∆ , where the last inequality is a consequence of the factsthat N ( a ) ∩ N ( u ) = ∅ and G does not contain a copy of K ,t . In summary, we have shown theimplication | A ∗ ( u ) | ≥ | N ( u ) | + | A ∗∗ ( u ) | = ⇒ | B ( u ) | ≤ ( t − · ∆ . (15)9 u N ( u ) ∩ ( N ( N ( u )) \ N ( N ( u ))) very small N ( u ) ∩ N ( N ( u ))) N ( N ( u ) ∩ N ( N ( u ))) small N ( N ( u ) ∩ ( N ( N ( u )) \ N ( N ( u )))) small small small small Figure 6: A depiction of (12) in Case ( iii ) of Theorem 1.1.Combining (14) and (15) yields our first desired main consequence (11).We now prove inequality (12), the second consequence. See Figure 6. First, the absence of bluecopies of K ,t implies that for every x ∈ N ( u ) we have | N ( x ) ∩ N ( u ) | ≤ t −
1. Therefore | N ( u ) ∩ N ( N ( u )) | ≤ | N ( u ) | · max x ∈ N ( u ) ( | N ( x ) ∩ N ( u ) | ) ≤ ∆ · ( t − . In other words, for at most ∆ · ( t −
1) vertices y ∈ N ( u ) there is a red–blue-link from u to y .Recalling that there is a link from u to every vertex (possibly with the exception of v ), it followsthat there are at least h := | N ( u ) | − ( t − − y ∈ N ( u ) for which there is a blue–red-link (and no red–blue-link) from u to y . In other words, m := | N ( u ) ∩ N ( N ( u )) | ≥ h .It follows from the definition of blue–red-link that any y ∈ N ( u ) ∩ N ( N ( u )) is connected toat least one other vertex y ∈ N ( u ) ∩ N ( N ( u )) by a red edge . If m is even, this means thatthere exists a matching of N ( u ) ∩ N ( N ( u )) consisting of red edges y y , . . . , y m − y m . Each ofthese edges has a large common red neighbourhood: for all i ∈ { , , , . . . , m − } it holds that | N ( y ) ∪ N ( y ) | = | N ( y ) | + | N ( y ) | − | N ( y ) ∩ N ( y ) | ≤ ∆ + ∆ − (1 − (cid:15) )∆ = (1 + (cid:15) )∆ . So | N ( N ( u ) ∩ N ( N ( u ))) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:91) y ∈ N ( u ) ∩ N ( N ( u )) N ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:91) i ∈{ , ,...,m − } | N ( y i ) ∪ N ( y i +1 ) | ≤ m · (1 + (cid:15) ) · ∆ . If, on the other hand, m is odd, then the same (or actually an even better) bound holds, becausethere exists a near-matching of N ( u ) ∩ N ( N ( u )) with red edges y y , . . . , y m − y m − and a red2-path consisting of edges y m − y m and y m − y m satisfying | N ( y m − ) ∪ N ( y m − ) ∪ N ( y m ) | ≤| N ( y m ) | + | N ( y m − ) \ N ( y m ) | + | N ( y m − ) \ N ( y m ) | ≤ (1 + 2 (cid:15) )∆ ≤ · (1 + (cid:15) ) · ∆ .Last, note that | N ( u ) ∩ ( N ( N ( u )) \ N ( N ( u ))) | = | N ( u ) | − m − { (cid:64) link from u to v } ≤ | N ( u ) | − m.
10e are now ready to derive (12): | N ( N ( u )) | ≤ | N ( N ( u ) ∩ N ( N ( u ))) | + | N ( N ( u ) ∩ ( N ( N ( u )) \ N ( N ( u )))) | + | N ( v ) |≤ m · (1 + (cid:15) ) · ∆ + ( | N ( u ) | − m ) · ∆ + ∆ =: g ( m ) . Since ∆ ≥ (cid:15) < /
2, the function g ( x ) is nonincreasing on the whole of R . Since h ≤ m , itfollows that g ( m ) ≤ g ( h ). So | N ( N ( u )) | ≤ g ( | N ( u ) | − ( t − − (cid:15) · ( | N ( u ) | − ( t − · ∆ − · ∆ + ( t − · ∆ + 2∆ ≤ (cid:15) · ∆ ∆ + 1 − (cid:15) · ( t − · ∆ + 3 − (cid:15) · ∆ , as desired.Finally, we evaluate (11) and (12) in the bounds on n given by Claim 3.3, parts ( i ) and ( ii ), toobtain n ≤ min ( | N ( N ( u )) | + | A ∗ ( u ) | + | N ( u ) | , | N ( N ( u )) | + | N ( u ) | + | N ∗ ( u ) | + | B ( u ) | ) ≤ min (cid:18) ∆ ∆ + ∆ + | A ∗ ( u ) | , (cid:15) ∆ + 1 − (cid:15) t − + (cid:18) t + 3 − (cid:15) (cid:19) · ∆ + | B ( u ) | (cid:19) = ∆ ∆ + ∆ + min (cid:18) | A ∗ ( u ) | , | B ( u ) | + (cid:18) t + 1 − (cid:15) (cid:19) · ∆ − − (cid:15) (cid:0) ∆ ∆ − ( t − (cid:1)(cid:19) = ∆ ∆ + ∆ + max (cid:18) + (cid:15) ∆ − (cid:15) , − (cid:15) t − · ∆ − − (cid:15) ∆ + (cid:18) t + 1 − (cid:15) (cid:19) · ∆ (cid:19) , (16)where we employed (11) and (12) only in the last line. Suppose the theorem is false. Consider a critical counterexample, a pair of non-packable graphs( G , G ) satisfying the constraints of the theorem, such that there is a near-packing with a uniquepurple edge uv . We distinguish two cases, Cases ( i ) and ( ii ). From the first we derive the inequal-ity (17) and from the second we obtain the inequality (18). Together they contradict the conditionthat max((5) , (6)) < n , thus proving the theorem.( i ) | A ∗ ( u ) | ≥ αt · ∆ (∆ + 1) or | A ∗ ( v ) | ≥ αt · ∆ (∆ + 1).Without loss of generality, we assume | A ∗ ( u ) | ≥ αt · ∆ (∆ + 1). From here the proof is the sameas for Case ( i ) in the proof of Theorem 1.1, leading to the same bound, n ≤ (cid:18) t + α ( α − α − − α (cid:19) · ∆ + ∆ ∆ . (17)( ii ) Case ( i ) does not hold.From here we proceed almost exactly as for Case ( ii ) in the proof of Theorem 1.1, the differencebeing that instead of the upper bound | N ( u ) ∩ N ( v ) | < (1 − (cid:15) ) · ∆ we use | N ( u ) ∩ N ( v ) | < s ,which holds due to the additional condition ∆ (cid:77) < s . (Compare with (10).) It follows that n ≤ αt · ∆ (∆ + 1) + 2∆ + ∆ · ( s −
1) + ∆ · ( t − . (18)11 .3 Concluding remarks We wish to make the following remarks about Theorems 1.1 and 1.2. • In Theorem 1.1, the bottleneck is the quantity (2), which corresponds to the bound (10) ofCase ( ii ). So improving in this case would improve the overall bound on n , albeit not by much. • The condition in Theorem 1.2 that ∆ (cid:77) < s is equivalent to “ | N ( x ) ∩ N ( y ) | < s for all xy ∈ E ( G )”. With a little adaptation, we can replace this by the weaker but perhaps obscurecondition that G has no subgraph G !2 such that | N ( x ) ∩ N ( y ) | ≥ s for all xy ∈ E ( G !2 ).Indeed, this property is invariant under edge removal, and so holds for an edge-minimal criticalcounterexample, which therefore has an edge uv with | N ( u ) ∩ N ( v ) | < s , for which we canchoose labellings such that uv is the unique purple edge. From here, one again proceeds exactlyas in Case ( ii ) of the proof of Theorem 1.1. • Theorem 1.2 yields a better bound than Theorem 1.1 only if ∆ is much larger than ∆ and s , t are both small. • By taking G to be a collection of (nearly) equal-sized cliques, Corollary 1.4 implies that, if G is a K ,t -free graph of maximum degree ∆ with ∆ ≥ √ t · √ n , then the equitable chromaticnumber of G is at most ∆. Note that this result cannot be obtained by the result of Hajnaland Szmer´edi on equitable colourings [10].The BEC conjecture notwithstanding, naturally one might wonder whether Theorem 1.1, or ratherCorollary 1.5, could be improved according to a weaker form of the BEC condition, as was thecase for d -degenerate G [4]. In other words, it would be interesting to improve upon the Ω(∆ ∆ )terms appearing in each of (1)–(4). We leave this to further study, but point out the followingconstructions where G has low maximum codegree, which mark boundaries for this problem. • When n is even, there are non-packable pairs ( G , G ) of graphs where G is a perfect matching(so ∆ ∧ = 0) and 2∆ ∆ = n , cf. [12]. • Bollob´as, Kostochka and Nakprasit [3] exhibited a family of non-packable pairs ( G , G ) ofgraphs where G is a forest (so ∆ ∧ = 1) and ∆ ln ∆ ≥ cn for some c > • If ∆ ∧ ( G ) = 1, then the chromatic number of G satisfies χ ( G ) = O (∆( G ) / ln ∆( G )) as ∆( G ) →∞ , and there are standard examples having arbitrarily large girth that show this bound tobe sharp up to a constant factor, cf. [14, Ex. 12.7]. Since the equitable chromatic number isat least the chromatic number, these examples moreover yield non-packable pairs ( G , G ) ofgraphs having ∆ ln ∆ (∆ + 1) ≥ cn for some c > ∧ = 1.Since the examples can also have the maximum adjacent codegree ∆ (cid:77) being zero, this last remarkhints at another natural line to pursue, which could significantly extend both the result of Csaba [8]and a result of Johansson [11]. If ∆ is large enough and G is triangle-free, is some condition ofthe form ∆ ln ∆ (∆ + 1) = cn for some constant c > G and G to pack? References [1] M. Aigner and S. Brandt. Embedding arbitrary graphs of maximum degree two.
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