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The Vertex Arboricity of Integer The Vertex Arboricity of Integer Distance Graph with a Special Distance Graph with a Special
Distance SetDistance Set
Juan Liu* and Qinglin Yu
Center for Combinatorics, LPMCNankai University, Tianjin 300071, P. R. China
OutlineOutline
Definitions and NotationsBackground and Known ResultsMain Theorem
Definitions and NotationsDefinitions and Notations
Vertex arboricity
Given a graph G, a k-coloring of G is a mapping from V(G) to [1, k].
denotes the set of all vertices of G colored with i, and denotes the subgraph induced by in G.
iV
iV
iV
Chromatic Number VS Vertex Arboricity Chromatic Number VS Vertex Arboricity
a proper k-coloring:
each is an
independent set. chromatic number
= min {k|G has a
proper k-coloring}
a tree k-coloring:
each induces a forest.
vertex arboricity va(G)
va(G) = min {k|G has a
tree k-coloring }
( )G
( )G
iV iV
Vertex ArboricityVertex Arboricity
Vertex arboricity is the minimum number of subsets into which V(G) can be partitioned so that each subset induces an acyclic subgraph of G.
Clearly,
for any graph G.( ) ( )va G G
ExamplesExamples
( ) 2va G =
( ) 5Gc =
5( ) 3va K =
(G) 3c =
( ) 3va G =
5(K ) 5c =
Known results for Known results for vava(G)(G)
(Kronk & Mitchem, 1975) For any graph G,
(Catlin & Lai, 1995) If G is neither a cycle nor a clique, then
( ) 1( )
2
Gva G
( )( )
2
Gva G
Known results for Known results for vava((GG))
(Skrekovski, 1975) For a locally planar
graph G, ; For a triangle-free
locally planar graph G, .
(Jorgensen, 2001) Every graph without a
-minor has vertex arboricity at most 4.
( ) 3va G ( ) 2va G
4,4K
Definitions and NotationsDefinitions and Notations
Distance graph If and , then the distance graph G(S, D) is defined by the graph with vertex set S and two vertices x and y are adjacent if and only if
where the set D is called the distance set.
S D
| |x y D
Definitions and NotationsDefinitions and Notations
Integer distance graph
if and all elements of D are positive integers, then the graph G(Z, D)=G(D) is called the integer distance graph and the set D is called its integer distance set.
S
Examples ofExamples of Integer Distance GraphInteger Distance Graph
D={2} -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
D={1, 3}
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
BackgroundBackground
The distance graph was introduced by Eggleton et al in 1985.
Coloring problems on distance graphs are motivated by the famous Hadwiger-Nelson coloring problem on the unit distance plane.
Known resultsKnown results
Chromatic number of integer distance graph;
Vertex arboricity of integer distance graph.
Results onResults on
(Eggleton, Erdos & Skilton, 1984)
where D is an interval
between 1 and for .
( ( , )) 2G R D n
1 1n n
( , )G R D
Results onResults on
(Eggleton, Erdos & Skilton, 1985) If a and b are relatively prime positive integers of opposite parity, then .
(Eggleton, Erdos & Skilton, 1986)
where P is the set of primes.
( , { , }) 3a b
( ) 4P
( , )G D
Results on Results on
(Chen, Chang & Huang, 1994) If
D={a,b,a+b}, where and
gcd{a,b}=1 then3, (mod 3)
( )4, (mod 3)
if a bD
if a b
1 a b
( , )G D
Integer Distance GraphInteger Distance Graph
m=4, k=2, D={1, 3, 4}
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
,( )m kG D
, [1, ] \{ }m kD m k=
Results on graphResults on graph
(Chang, Liu & Zhu, 1999) Let then
, [1, ] \{ }m kD m k
,( )m kG D
,
1
2( , )
1,
2
m k
m kif r s
Dm k
otherwise
Vertex arboricity ofVertex arboricity of
(Yu, Zuo & Wu) For any integer
,( )m kG D
5m
,1
3( ( ))
4m
mva G D
Vertex arboricity ofVertex arboricity of
(Yu, Zuo & Wu) Let with
for a positive integer ,
we have
,( )m kG D
6m
,2
11 7
4( ( ))
1 2 74 4
m
mfor j
va G Dm m
or for j
8 6m l j 0 8j
Vertex arboricity ofVertex arboricity of
(Yu, Zuo & Wu) For any
with , we have
4 3 9m kl j k 0 4j k
,
1 2 1( ( ))
4 4m k
m k m kva G D k
k
,( )m kG D
Main theoremMain theorem
Theorem for
Proof (1) Upper bound
(2) Lower bound
,2( ( )) 24m
mva G D
8 7 15m l
(1) Upper bound(1) Upper bound
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 8(l+2) 8(l+2)+6 8(l+2)+14
(2) Lower bound(2) Lower bound
Lemma (Shifting Lemma) Let , be subgraphs of G(D) induced by vertices
and vertices for any
respectively. Then has a tree n-coloring if and only if has a tree n-coloring.
1H 2H
[ , ] ( , )c l c l Î
s Î
[ , ]c s l s+ +
1H
2H
(2) Lower bound(2) Lower bound
By contradiction. Assume, on the
contrary, that
then has a tree (2l+3)-coloring f.
,2( ( )) 1 2 34m
mva G D l
,2( )mG D
(2) Lower bound(2) Lower bound
Find a finite subgraph H and try to get a
contradiction.
Question: How to find such a subgraph H?
How to get a contradiction?
How to find such a subgraphHow to find such a subgraph HH??
By hypothesis, f is also a tree (2l+3)-
coloring of H.
● We consider a subgraph H induced by
the vertex subset [0, m+5].
How to get a contradiction inHow to get a contradiction in HH??
Note that |V(H)|=m+6. There exist at least
five vertices in H, say ,
which are colored by the same color .
Question: Can we prove that any color except
just colors four vertices?
Answer: Yes!
0 1 40 5a a a m
How to prove?How to prove?
There are at most five vertices receiving the color in H.
There isn't any other color, except , coloring five vertices in H.
There are at most five verticesThere are at most five verticesreceiving the color in receiving the color in HH..By contradiction. Suppose the color
colors six vertices in H.
Need only to consider the case when
because we can shift the interval to [-1, m+4] when , and get a contradiction similarly in H.
0 1 50 4a a a m
5 5a m 5 4a m= +
There isn't any other color, except There isn't any other color, except coloring five vertices in coloring five vertices in HH..
By contradiction. Note that m=8l+7, use the divisibility of m.
Get a contradiction in Get a contradiction in HH
Main idea: Find a vertex whose coloring will result in a cycle in some color set.
Get information about the location of the five vertices as much as possible.
Make cases needed to consider as few as possible.
Use vertices as few as possible.
Thank you!Thank you!
