A Coloring Algorithm for Triangle-Free Graphs

Abstract

We give a randomized algorithm that properly colors the vertices of a
triangle-free graph G on n vertices using O(\Delta(G)/ log \Delta(G)) colors,
where \Delta(G) is the maximum degree of G. The algorithm takes
O(n\Delta2(G)log\Delta(G)) time and succeeds with high probability, provided
\Delta(G) is greater than log^{1+{\epsilon}}n for a positive constant
{\epsilon}. The number of colors is best possible up to a constant factor for
triangle-free graphs. As a result this gives an algorithmic proof for a sharp
upper bound of the chromatic number of a triangle-free graph, the existence of
which was previously established by Kim and Johansson respectively.

Full text

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arXiv:1101.5721v1 [math.CO] 29 Jan 2011
A Coloring Algorithm for Triangle-Free Graphs
Mohammad Shoaib Jamall⋆
Department of Mathematics, UC San Diego
mjamall@math.ucsd.edu
Abstract. We give a randomized algorithm that properly colors the vertices of a triangle-free graph G
on n vertices using O(∆(G)/log∆(G)) colors, where ∆(G) is the maximum degree of G. The algorithm
takes O(n∆2(G)log∆(G)) time and succeeds with high probability, provided ∆(G) is greater than
log1+ǫn for a positive constant ǫ. The number of colors is best possible up to a constant factor for
triangle-free graphs. As a result this gives an algorithmic proof for a sharp upper bound of the chromatic
number of a triangle-free graph, the existence of which was previously established by Kim and Johansson
respectively.
1 Introduction
A proper vertex coloring of a graph is an assignment of colors to all vertices such that adjacent vertices have
distinct colors. The chromatic number χ(G) of a graph G is the minimum number of colors required for
a proper vertex coloring. Finding the chromatic number of a graph is NP-Hard [10]. Approximating it to
within a polynomial ratio is also hard [15]. For general graphs, ∆(G) + 1 is a trivial upper bound. Brooks’
Theorem [7] shows that χ(G) can be ∆(G)+1 only if G has a component which is either a complete subgraph
or an odd cycle.
A natural question is: can this bound be improved for graphs without large complete subgraphs? In
1968, Vizing [22] had asked what the best possible upper bound for the chromatic number of a triangle-free
graph was. Borodin and Kostochka [6], Catalin [8], and Lawrence [19] independently made progress in this
direction; they showed that for a K4-free graph, χ(G) ≤ 3(∆(G) + 2)/4. On the other hand, Kostochka and
Masurova [18], and Bollob´ as [5] separately showed that there are graphs of arbitrarily large girth(length of
a shortest cycle) with χ(G) of order ∆(G)/log∆(G).
Further progress was made using the semi-random method to show that the chromatic number of graphs
with large girth is O(∆(G)/log∆(G)). This technique, also known as the pseudo-random method, or the
R¨ odl nibble, appeared first in Ajtai, Koml´ os and Szemer´ edi [2] and was applied to problems in hypergraph
packings, Ramsey theory, colorings, and list colorings [9,13,14,17,21]. In general, given a set S1, the goal
is to show that there is an object in S1with a desired property P. This is done by locating a sequence of
non-empty subsets S1 ⊇ ··· ⊇ Sτ with Sτ having property P. A randomized algorithm is applied to St,
which guarantees that St+1will be obtained with some non-zero(often small) probability. For upper bounds
on chromatic number, the semi-random method is used to prove the existence of a proper coloring with a
limited number of colors. It does not give an efficient probabilistic algorithm.
In 1995, Kim [16] proved that
χ(G) ≤ (1 + o(1))
∆(G)
log∆(G)
when G has girth greater than 4. Later on, Johansson [12] showed that
χ(G) ≤ O(
∆(G)
log∆(G))
when G is a triangle-free graph(girth greater than 3). Both Kim and Johansson used the semi-random
method. Alon, Krivelevich and Sudakov [3], and Vu [23] extended the method of Johansson to prove bounds
⋆Research supported in part by a Reuben H. Fleet Foundation Fellowship, and an ARCS Foundation Fellowship.

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on the chromatic number for graphs in which no subgraph on the set of all neighbors of a vertex has too
many edges.
Grable and Panconesi [11] gave an algorithm for properly coloring a ∆-regular graph with girth greater
than 4 and ∆ ≥ log1+ǫn for a positive constant ǫ. The procedure uses O(∆/log∆) colors and has polynomial
running time. This is a constructive version of the existential proof of Kim with extra conditions on the degree
of the input graph. Section 5 of [11] asserts that the algorithm can be extended to triangle-free graphs. In
Section 2.1 we will see a counter-example that shows that their analysis does not work on all triangle-free
graphs. In particular, as will be seen, the analysis fails on a complete bipartite graph.
In this paper we give a method for coloring the larger class of triangle-free graphs. Our randomized
algorithm properly colors a triangle-free graph G on n vertices with ∆(G) ≥ log1+ǫn for a positive constant
ǫ, using O(∆(G)/log∆(G)) colors in O(n∆2(G)log∆(G)) time. The probability of failure goes to 0 as n
becomes large.
To analyze our iterative algorithm we identify a collection of random variables. The expected changes
to these random variables after a round of the algorithm are written in terms of the values of the random
variables before the round. We thus obtain a set of recurrence relations and prove that our random variables
are concentrated around the solutions to the recurrence relations.
Our method of analysis resembles that of Kim, and Grable et al. However our algorithm and collection
of random variables are different. Also related to our analysis technique is Achlioptas and Molloy’s study of
the performance of a coloring algorithm on random graphs [1]. They setup recurrence relations as we do,
and then use the so-called differential equation method for random graph processes [24].
We describe our algorithm in Section 2. Section 2.1 contains motivation, which is followed by a formal
description of the algorithm in Section 2.2. We give an outline of the analysis in Section 3. Section 4 contains
some useful lemmas which we are used in Section 5 to give details of the analysis.
2 An Iterative Algorithm for Coloring a Graph
Our algorithm takes as input a triangle-free graph G on n vertices, its maximum degree ∆, and the number
of colors to use ∆/k where k is a positive number. It goes through rounds and assigns colors to more vertices
each round. Initially all vertices are uncolored(no color assigned), at the end we have a proper vertex coloring
of G with some probability.
Definition 1. Let t be a natural number. We define the following:
Gt
The graph induced on G by the vertices that are uncolored at the beginning of round t.
Nt(u) The set of vertices adjacent to vertex u in Gt. That is, the set of uncolored neighbors of u
at the beginning of round t.
St(u) The list of colors that may be assigned to vertex u in round t, also called the palette of u.
For all u in V (G),
S0(u) = {1,...,∆/k}.
Dt(u,c) The set of vertices adjacent to u that may be assigned color c in round t. That is,
Dt(u,c) := {v ∈ Nt(u)|c ∈ St(v)}.
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It will be useful to define variables for the sizes of the sets Nt(u), St(u), and Dt(u,c).
Definition 2.
ηt(u) = |Nt(u)|
st(u) = |St(u)|
dt(u,c) = |Dt(u,c)|
Observe that for every round t, vertex u in V (G), and color c in {1,...,∆/k},
dt(u,c),ηt(u) ≤ ∆, s0(u) = ∆/k, st(u) ≤ ∆/k.
2.1 A Sketch of the Algorithm and the Ideas behind its Analysis
We start by considering an algorithm that works for ∆-regular graphs with girth greater than 4. The rounds
of this algorithm will go through two stages. The number of colors in the palette of each uncolored vertex
will be greater than the number of uncolored adjacent vertices at the end of the second stage with some
probability. Then the partial coloring of the graph can be completed easily to give a proper coloring.
Let (ηt), (dt), and (st) be sequences defined recursively as
η0:= ∆
d0:= ∆
s0:= ∆/k
ηt+1:= ηt(1 − c1st
dt+1:= dt(1 − c1st
st+1:= stc2
dt)
dt)c2
where c1and c2are constants between 0 and 1, which are determined by the analysis of the algorithm.
First Stage
Repeat at every round t, until dt/st< 1
Phase I - Coloring Attempt
For each vertex u in Gt:
Awake vertex u with probabilitity st/dt.
If awake, assign to u a color chosen from St(u) uniformly at random.
Phase II - Conflict Resolution
For each vertex u in Gt:
If u is awake,
uncolor u if an adjacent vertex is assigned the same color in the coloring attempt phase.
Remove from St(u), all colors assigned to adjacent vertices.
St+1(u) = St(u).
end repeat
Observe that d0/s0= k and
dt+1
st+1
=dt
st
− c1.
In O(k) rounds dt/stwill be less that 1, and this marks the end of the first stage.
Second Stage We change the recursive equations for ηt, dt, and st.
ηt+1:= ηt(1 − c1ec2dt/st)
dt+1:= dt(1 − c1ec2dt/st)e−c2dt/st
st+1:= ste−c2dt/st
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All uncolored vertices are woken up at every round. In this stage, ηtdecreases much faster than stand
the repeat-until loop is repeated until
ηt< st.
Otherwise the second stage is the same as the first.
The two stage algorithm is derived from Grable et al. Their analysis tells us that if graph G has girth greater
than 4 and ∆ ≥ log1+ǫn for some positive constant ǫ, then there are constants c1and c2less than 1 such
that at each round t, ∀u ∈ V (Gt),∀c ∈ St(u)
ηt(u) = ηt(1 + o(1)),st(u) = st(1 + o(1)),dt(u,c) = dt(1 + o(1)) (1)
with probability 1 − o(1). The equations above imply that at the end of the second stage st(u) > ηt(u) for
all uncolored vertices u with probability approaching 1 as n approaching ∞.
The change we have made to Grable et al. so far, is that we remove all colors temporarily assigned to
neighbors of a vertex from its palette, instead of removing only those assigned permanently. This simple but
powerful idea, adapted from Kim [16], will waste a few colors from the palettes, and simplify our algorithm
and its analysis significantly.
A counter-example to demonstrate that Grable et al. does not work on graphs with girth
greater than 3. The analysis for the above algorithm is probabilistic and proves the property in equation
(1) by induction, showing concentration of the variables around their expectations. It fails for graphs with
4-cycles. An example illustrates why: Consider a vertex u whose 2-neighborhood, the graph induced by
vertices within distance 2 of u, is the complete bipartite graph K∆,∆ with partitions X and Y . Suppose
that u and another vertex v are in X. If v is colored with c in round 0 while u remains uncolored, then the
set D1(u,c) = ∅; this violates equation (1) since d1≥ 1 if for example k ≥ 2 and ∆ ≥ 2/c2. So, when the
graph has 4-cycles, dt+1(u,c) is not necessarily concentrated around its expectation given the state of the
algorithm at the beginning of round t. Maintaining equation (1) is crucial for the proof in Grable et al., and
this violation is the error we mentioned in the introduction; their algorithm and analysis do not work for
triangle-free graphs in general!
We must modify the algorithm in two more ways.
First Modification: A technique for coloring graphs with 4-cycles. While dt(u,c) is not concentrated
enough when the graph has 4-cycles, our analysis will show that the average of dt+1(u,c) over all colors
in the palette of a vertex u is concentrated enough. How does this benefit us? Markov’s famous inequality
may be interpreted as: a list of s positive number which average d has at most s/q numbers larger than
qd for any positive number q. We modify the algorithm so that at the end of each round t, every vertex u
removes from its palette every color c with dt+1(u,c) larger than qdt+1for some constant q larger than 1.
Look at what happens in round t = 1. By a straightforward application of Markov’s inequality, instead
of equation (1) we will have the less stringent property: ∀u ∈ V (Gt),∀c ∈ St(u)
ηt(u) ≤ ηt(1 + o(1)), st(u) ≥q − 1
q
st(1 − o(1)), dt(u,c) ≤ qdt(1 + o(1)). (2)
with probability 1 − o(1). In fact, using a generalization of Markov’s inequality, the analysis will show
that with a few more modifications our algorithm maintains a slightly stronger property(still weaker
than equation (1)).
Equation (1) implies that the ηt(u), st(u), and dt(u) at all uncolored vertices u are about the same.
It is a strong statement and helps in the proofs, but is too much to maintain on graphs with 4-cycles.
Equation (2) is weaker and is guaranteed by our algorithm and what is more, it is sufficient to guarantee
that all uncolored vertices at the end of the second stage have palettes with more colors than the number
of uncolored adjacent vertices. This is a key idea in our algorithm.
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Second Modification: Using independent random variables for easier analysis. Instead of waking
up a vertex with some probability, and then choosing a color from its palette uniformly at random; for
each uncolored vertex u and color c in its palette, we will assign c to u independently with some prob-
ability. In case multiple colors remain assigned to the vertex after the conflict resolution phase, we
will arbitrarily choose one of them to permanently color the vertex. This modification, adapted from
Johansson [12], will make concentration of our random variables simpler.
Next we provide a formal description of the algorithm we have just motivated.
2.2 A Formal Description of the Algorithm
In each round of the algorithm, some vertices are colored. The details of the coloringprocedure vary depending
on which of three stages the algorithm is in.
First Stage Let q be a constant greater than 1, and let (ηt), (dt), and (st) be sequences defined recursively
as
η0:= ∆ηt+1:= ηt(1 −q − 1
2q3e−1/qst
2q3e−1/qst
dt)
d0:= ∆dt+1:= dt(1 −q − 1
dt)e−1/q
s0:= ∆/kst+1:= ste−1/q.
(3)
For round t, vertex u, and color c,
Ft(u,c) := {c is not assigned to any vertex adjacent to u in round t}
(4)
is an event in the probability space generated by the random choices of the algorithm in round t, given
the state of all data structures at the beginning of the round.
Let
Desired Ft:= e−1/q.
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