Conflict-Free Coloring of Star-Free Graphs on Open Neighborhoods
Sriram Bhyravarapu, Subrahmanyam Kalyanasundaram, Rogers Mathew
aa r X i v : . [ m a t h . C O ] S e p Conflict-Free Coloring of Star-Free Graphs onOpen Neighborhoods
Sriram Bhyravarapu, Subrahmanyam Kalyanasundaram, andRogers MathewDepartment of Computer Science and Engineering,Indian Institute of Technology Hyderabad, India - 502285. { cs16resch11001, subruk, rogers } @iith.ac.in September 16, 2020
Abstract
Given a graph, the conflict-free coloring problem on open neighbor-hoods (CFON) asks to color the vertices of the graph so that all thevertices have a uniquely colored vertex in its open neighborhood. Thesmallest number of colors required for such a coloring is called the conflict-free chromatic number and denoted χ ON ( G ). In this note, we study thisproblem on S k -free graphs where S k is a star on k + 1 vertices. When G is S k -free, we show that χ ON ( G ) = O ( k · log ǫ ∆), for any ǫ > G . Further, we show existenceof claw-free ( S -free) graphs that require Ω(log ∆) colors. A conflict-free coloring of a hypergraph H = ( V, E ), denoted by χ CF ( H ),is an assignment of colors to the points in V such that every e ∈ E contains apoint whose color is distinct from that of every other point in e . The conflict-free chromatic number of H , denoted χ CF ( H ), is the smallest number of colorsrequired for such a coloring. Introduced [ELRS04] in 2002 by Even, Lotker,Ron and Smorodinsky, many variants of the problem have been extensivelystudied [Smo13]. Conflict-free colorings have also been studied in the contextof hypergraphs created out of graphs. One such popular variant is conflict-freecoloring with respect to open neighborhoods (or CFON coloring) in a graph.Given a graph G , for any vertex v ∈ V ( G ), let N G ( v ) := { u ∈ V ( G ) : { u, v } ∈ E ( G ) } denote the open neighborhood of v in G . Definition 1 (CFON Coloring of Graphs) . Given a graph G , let H be the hy-pergraph with V ( H ) = V ( G ) and E ( H ) = { N G ( v ) : v ∈ V ( G ) } . A conflict-freecoloring on open neighborhoods (CFON coloring) of G is defined as a conflict-free coloring of H . The CFON chromatic number of G , denoted by χ ON ( G ) , isequal to χ CF ( H ) . efinition 2 (Star and Claw) . The complete bipartite graph K ,k is referred toas star on k + 1 vertices and denoted by S k . The graph S is also known by thename claw . A graph is said to be S k -free ( claw-free ) if it does not contain an S k ( S ) as aninduced subgraph. In this paper, we study the CFON problem on S k -free graphs.For a graph G with maximum degree ∆, it is known that χ ON ( G ) ≤ ∆ + 1 andthis bound is tight in general. We improve this result for S k -free graphs formost values of k . Theorem 3.
Let G be an S k -free graph with maximum degree ∆ . Then, χ ON ( G ) = O ( k log ǫ ∆) , for any ǫ > . The following Theorems 4 and 5 from [PT09] will be used in the proof ofTheorem 3.
Theorem 4 (Theorem 1.1 in [PT09]) . Let H be a hypergraph and let ∆ be themaximum degree of a vertex in H . Then the conflict-free chromatic number of H is at most ∆ + 1 . This bound is optimal and the corresponding coloring canbe found in linear deterministic time. Let G be a graph with maximum degree ∆. The above theorem implies that χ ON ( G ) ≤ ∆ + 1. The subdivided clique K ∗ n has a maximum degree of n − χ ON ( K ∗ n ) = n . This serves as a tight example to the above bound. Theorem 5 (Theorem 1.2 in [PT09]) . For any positive integers t and Γ , theconflict-free chromatic number of any hypergraph in which each edge is of sizeat least t − and each edge intersects at most Γ others is O ( t Γ /t log Γ) . Thereis a randomized polynomial time algorithm to find such a coloring. We begin with an auxiliary lemma.
Lemma 6.
Let G be an S k -free graph with no isolated vertices. Let A ⊆ V ( G ) be an independent set of vertices in G . Let B = V ( G ) \ A . There is a way tocolor the vertices in B using at most k colors such that every vertex in A seessome color appear exactly once in its open neighborhood.Proof. Construct a hypergraph H = ( V, E ) with V ( H ) = B and E ( H ) = { N G ( v ) ∩ B : v ∈ A } . Since G is S k -free, no element of V ( H ) is presentin more than k − H using k colors. Proof of Theorem 3.
We use an iterative process to color the vertices. Consideran S k -free graph G = ( V, E ) with maximum degree ∆ = ∆( G ). We first par-tition V = V into U and V , where U = { v ∈ V : deg G [ V ] ( v ) > log ∆ } ,and V = V \ U . We construct a hypergraph H from G with V ( H ) = V and E ( H ) = { N G ( v ) ∩ V : v ∈ U } . Every hyperedge e ∈ E ( H ) satisfies | e | ≥ log ∆ + 1, and e intersects at most ∆ other hyperedges in H . ApplyingTheorem 5 with t = log ∆2 + 1 and Γ = ∆ , we get the conflict-free chromatic2umber of H to be at most α (log ∆) , where α > C : V → [ α (log ∆) ] such that every vertex in U sees some color exactly once in its open neighborhood.Now notice that V is the set of all vertices that have degree at mostlog ∆. We repeat the above process by setting U = { v ∈ V : deg G [ V ] ( v ) > log ∆( G [ V ]) ≥ log log ∆ } , and V = V \ U . We construct a hypergraph H with V ( H ) = V and E ( H ) = { N G ( v ) ∩ V : v ∈ U } . Every hyperedge in H is of size at least log log ∆ + 1 and each hyperedge intersects at most log ∆other hyperedges. Applying Theorem 5 with t = log log ∆2 + 1 and Γ = log ∆,we get an assignment C : V → [ α (log log ∆) ] such that every vertex in U sees some color exactly once in its open neighborhood.We iterate in this manner till we get, say V r , which is an independent set in G . We now use Lemma 6 to assign C r +1 : V → [ k ] so that every vertex in V r sees a color exactly once in its open neighborhood.Now consider the color assignment C formed by the Cartesian product ofthe previous assignments. That is C ( v ) = ( C ( v ) , C ( v ) , . . . , C r ( v ) , C r +1 ( v )).Notice that C is a CFON coloring. This is because V = U ∪ U ∪ . . . ∪ U r ∪ V r is a partition of V . If v ∈ U i , then v has a neighbor that is uniquely colored bythe assignment C i . Also, every v ∈ V r has a neighbor that is uniquely coloredby the assignment C r +1 . The number of colors used is( α (log ∆) ) · ( α (log log ∆) ) · · · · · ( α (log log . . . log | {z } r times ∆) ) · k , which is upper bounded by k log ǫ ∆. This follows by noting that r ≤ log ∗ ∆,the iterated logarithm of ∆. Thus we have χ ON ( G ) = O ( k log ǫ ∆). Algorithmic note:
It is easy to see that the construction of sets V i and U i can be done in deterministic polynomial time. Theorem 4 states that the col-oring C r +1 can be computed in deterministic linear time. What is left is toknow whether the colorings C i (1 ≤ i ≤ r ) can be computed in deterministicpolynomial time. Theorem 5 states that the colorings C i (1 ≤ i ≤ r ) can beobtained in randomized polynomial time. In the proof of Theorem 5 in [PT09],an algorithmic version of the Local Lemma is used to obtain a randomized al-gorithm for finding the desired coloring for the hypergraph under consideration.There are deterministic algorithms known for the Local Lemma [CGH13, Har19]which can be used in place of the randomized algorithm used in [PT09]. By ap-plying Theorem 1.1 (1) from [Har19], we get a deterministic polynomial timealgorithm to find the colorings C i (1 ≤ i ≤ r ). However, the deterministicversion of Local Lemma causes us to use O ( t Γ (1+ δ ) /t log Γ) colors for a constant δ >
0. This is slightly worse than the bound in Theorem 5. However, this weakerbound suffices to get a conflict-free coloring of the hypergraphs H , H , . . . usingasymptotically the same number of colors as before. We thus have a determin-istic polynomial time algorithm for CFON coloring the vertices of an S k -freegraph with maximum degree ∆ using O ( k log ǫ ∆) colors, for any ǫ > Values C i ( v ) that are not assigned are notionally set to 0. G , the line graph of G , denoted by L ( G ), is the graph with V ( L ( G )) = E ( G ) and E ( L ( G )) = {{ e, f } : edges e and f share an endpoint in G } .It is easy to see that the line graph of any graph is claw-free. In what follows, weuse this fact to show the existence of claw-free graphs of high CFON chromaticnumber. Theorem 7.
There exist claw-free graphs G on n vertices with χ ON ( G ) =Ω(log n ) .Proof. Let m be a positive integer. Consider the complete graph K m on m vertices. Let n = (cid:0) m (cid:1) denote the number of vertices in the line graph of K m .Consider the CFON coloring problem for the line graph of K m . In other words,we need to color the edges of K m with the minimum number of colors such thatevery edge sees some color exactly once in its open neighborhood. Consider anoptimal CFON coloring C : E ( K m ) → { , , . . . , k } of the edges of K m thatuses, say k colors. Below we show that k ≥ log m .Corresponding to each v ∈ K m , we construct a k -bit 0-1 vector g ( v ). The i -th bit g i ( v ) = 1 if there is exactly one edge incident on v with the color i .Otherwise, g i ( v ) = 0. Since C is a valid CFON coloring of the edges of K m ,for any two distinct vertices u, v ∈ V ( K m ), g ( u ) should differ from g ( v ) in atleast one position. Consider the edge { u, v } ∈ E ( G ). Let { v, w } ∈ E ( G ) be theuniquely colored edge in the open neighborhood of { u, v } and let C ( { v, w } ) = i .This implies that none of the other edges incident on the vertices u or v areassigned the color i except possibly the edge { u, v } itself. Thus g ( u ) and g ( v )differ in at least one position. This implies that k ≥ log m .Since a line graph is claw-free, Theorems 3 and 7 imply the following corol-lary. Corollary 8.
Let G be the line graph of a graph. Let ∆ denote the maximumdegree of G . Then, χ ON ( G ) = O (log ǫ ∆) , for any ǫ > . Further, there existline graphs with maximum degree ∆ having χ ON ( G ) = Ω(log ∆) . Note:
Very recently, D¸ebski and Przyby lo [DP20], in independent and simulta-neous work, showed that the closed neighborhood conflict-free chromatic numberor CFCN chromatic number (defined analogously to Definition 1) of line graphsis O (log ∆). In Theorem 3 of [DP20], it is shown that χ CN ( L ( K n )) = Ω(log n ).Since the CFCN chromatic number of a graph is at most twice its CFON chro-matic number, this lower bound proved in [DP20] implies Theorem 7. References [CGH13] Karthekeyan Chandrasekaran, Navin Goyal, and Bernhard Haeupler.Deterministic algorithms for the Lov´asz local lemma.
SIAM Journalon Computing , 42(6):2132–2155, 2013.4DP20] Micha l D¸ebski and Jakub Przyby lo. Conflict-free chromatic numbervs conflict-free chromatic index. arXiv preprint arXiv:2009.02239 ,2020.[ELRS04] Guy Even, Zvi Lotker, Dana Ron, and Shakhar Smorodinsky.Conflict-free colorings of simple geometric regions with applicationsto frequency assignment in cellular networks.
SIAM Journal on Com-puting , 33(1):94–136, January 2004.[Har19] David G Harris. Deterministic algorithms for the Lov´asz locallemma: simpler, more general, and more parallel. arXiv preprintarXiv:1909.08065 , 2019.[PT09] Janos Pach and G´abor Tardos. Conflict-free colourings of graphs andhypergraphs.
Combinatorics, Probability and Computing , 18(5):819–834, 2009.[Smo13] Shakhar Smorodinsky.