2-rankings of Graphs

2-rankings of Graphs
Jordan Almeter, Samet Demircan, Andrew Kallmeyer
September 25, 2015
Jordan Almeter, Samet Demircan, Andrew Kallmeyer 2-rankings of Graphs September 25, 2015 1 / 17

Definitions
Definition
A ranking of a graph G is a proper coloring c : V (G) → [t] such that
∀u, v ∈ V (G) if P is a nontrivial uv-path with c(u) = c(v), then P
contains some vertex w where c(w) > c(u).
Definition
A k-ranking of a graph G is a proper coloring c : V (G) → [t] such that
∀u, v ∈ V (G) if P is a nontrivial uv-path of length at most k with
c(u) = c(v), then P contains some vertex w where c(w) > c(u).
Definition
The k-ranking-number of a graph G, denoted χk(G), is the minimum t
such that G has a k-ranking c : V (G) → [t].

2-ranking examples
1
2
2
2
1
1
1
3
2
good good bad

Background
Deﬁnition
A star coloring is a proper coloring of a graph G such that every path on
four vertices is colored with at least 3 colors. The star coloring number
of a graph is written as χS (G).
Deﬁnition
If G is a graph then G2 is the graph obtained from G by adding an edge
between every pair of vertices at distance two.
Observation
χ(G) ≤ χS (G) ≤ χ2(G) ≤ χ(G2)

Findings
2-Ranking of the Hypercube Qn
Applications related to Toroidal grids
2-Ranking of Cartesian products of complete graphs, Km Kn
2-Ranking of Subcubic Graphs
A probabilistic construction

2-Ranking of the Hypercube
Deﬁnition
G is a d-degenerate graph if every subgraph contains a vertex of degree
at most d
Lemma
Let G be a d-degenerate graph, then χ2(G) ≥ d + 1
Proof.
G must have a subgraph H with minimum degree d. Let v ∈ V (H) be a
vertex with the lowest rank used in H. Then, the neighbors of v must
have distinct colors and v has at least d neighbors, so χ2(H) ≥ d + 1.
Thus, χ2(G) ≥ d + 1.

2-Ranking of the Hypercube (cont.)
Theorem
For all hypercubes Qn on n dimensions, χ2(Qn) = n + 1
We prove this by construction.
Since Qn is n-degenerate, by the previous lemma we have
χ2(Qn) ≥ n + 1.
For all k if n = 2k − 1 then there is a coloring of Q2
n using n + 1
colors based on error correcting codes given by Ngo et al. (2002).
We extend this coloring to obtain 2-rankings for all Qn by adding
small-valued ”buffer” colors to this distance two coloring so that the
only paths of length two where the endpoints have the same rank are
ones with buffer colors at the endpoints.
An inductive coloring ensures that interactions between buffer
characters are valid.

2-Ranking of the Hypercube (cont.)
Each vertex x of a n-dimensional unit cube can be represented as a binary
bit string of length n which our coloring is based on. Speciﬁcally our
coloring of Qn, where n = 2k + t, is the following:
cn(x) =
ct([x1, . . . , xt]) if xt+1 + · · · + xn ≡ 0 mod 2
Ax + t otherwise
Where A is an augmented matrix with n columns and k rows. Within the
ﬁrst t columns we make all columns distinct and include no zero column.
The rightmost 2k columns are all of the distinct binary vectors of length k.
For example a valid matrix for Q6 is:
A =
1 0 1 0 1 0
0 1 0 1 1 0

2-Ranking of the toroidal grid
Deﬁnition
A d-dimensional toroidal grid is the cartesian product of d cycles.
Corollary (Hypercubes)
If G = Cn1 Cn2 · · · Cnd
and 4 divides each ni , then
χs(G) ≤ χ2(G) = 2d + 1
Theorem (Fertin, Raspaud, Reed)
If G = Cn1 Cn2 · · · Cnd
, then
d + 2 ≤ χs(G) ≤
2d + 1 when 2d + 1 divides each ni
2d2 + d + 1 otherwise

2-Ranking of Km Kn
This graph can be modeled as an m × n grid where vertices are adjacent if
they share a row or column. We may assume m ≤ n.
Lemma
In a 2-ranking of Km Kn, for all k ≤ m each column of height m must
contain k labels which appear k or fewer times in the graph.
Proof.
If the highest rank in a column appears more than once in the grid, we
have a high-low-high path; if the second highest rank appears more than
twice in the grid, we have a bad path; and so on.
This puts a limit on how many times a label can appear in a grid. For an
m × n grid, a ranking in which n labels appear only once, n/2 labels appear
only twice, n/3 labels appear only three times, and so on, is optimal.

2-Ranking of Km Kn (cont.)
From this argument, we can say
Theorem
If n ≥ m, then χ2(Km Kn) ≥ nHm, where Hm = 1 + 1
2 + 1
3 + · · · + 1
m .
In fact, if n is a multiple of m!, then equality holds.

For small values of n, χ2(Kn Kn) has been found, such as the following:
K2 K2 :
0 1
2 0
K3 K3 :
3 1 0
1 4 2
0 2 5
Through block matrix operations, using 2-ranking for Kn Kn we can ﬁnd
a 2-ranking for K2n K2n.
K6 K6 :
3 1 0 9 7 6
1 4 2 7 10 8
0 2 5 6 8 11
15 13 12 3 1 0
13 16 14 1 4 2
12 14 17 0 2 5

This construction yields the following result:
Theorem
χ2(Kmn Kmn) ≤ χ2(Km Km) · χ2(Kn Kn)
By recursively finding constructions for powers of two, we find the
following:
Theorem
If n is a power of 2, then χ2(Kn Kn) ≤ nlog2(3) ≈ n1.585.
This is the best upper bound we have been able to find so far.

Definition
A subcubic graph is a graph where each vertex has at most 3 neighbors.
Theorem
If G is subcubic, then χ2(G) ≤ 7
Lemma
For a subcubic graph G with maximal independent vertex set S, if
G2 − S = K7, then χ2(G) ≤ 7
Proof: color members of S with a buffer character 0, and then we find that
∆(G2 − S) ≤ 6 since each vertex in G − S must be adjacent to a vertex in
S. Then by Brooks’s theorem χ(G2 − S) ≤ 6 unless G2 − S contains K7.
Through case analysis, we find that the only graphs that have this
property are the Heawood graph and the Petersen graph, each of which
has a 2-ranking with 5 ranks.

Theorem (Chen, Raspaud, and Wang [2011])
χS (G) ≤ 6 for all 3-regular graphs
Conjecture
χ2(G) ≤ 6 for all subcubic graphs
In addition, we have only been able to ﬁnd one graph
which requires 6 colors; all other examples we have
found require at most 5 colors.

Further Results: Probabilistic Bound
Recall
χ2(G) ≤ χ(G2) ≤ ∆(G)2 + 1
Theorem
There exists a graph G with ∆(G) ≤ k and χ2(G) = Ω k2
log(k)
Proof:
Construct a random graph G(n, p) choosing n and p so that with
positive probability ∆(G) ≤ 2np and χ2(G) ≥ n
2
For each function f : V (G) → [n/2], let Af be the bad event that f is
a 2-ranking.
We have that P(χ2(G) ≤ n
2 ) = P( f Af ) < f P(Af )
So if P(Af ) < 1
(n/2)n we have that P(χ2(G) ≤ n
2 ) < 1 which is what
we want.

Further Results: Probabilistic Bound (cont.)
For f : V (G) → [n/2] we ﬁnd at least n/4 same-color pairs
u
vw
For pairs S and T the probability that f fails to be a 2-ranking
between them is at least p2
So we have that P(Af ) ≤ (1 − p2)(n/4
2 )
Solving (1 − p2)(n/4
2 ) < 1
(n/2)n we get p > c log n
n
1/2
With ∆(G) ≤ 2np = k we compute χ2(G) > n/2 = c k2
log k for some
positive constant c

2-rankings of Graphs

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