On parsimonious edge-colouring of graphs with maximum degree three

0.18 MB·English

by  Jean-Luc Fouquet

On parsimonious edge-colouring of graphs with
maximum degree three
J.L. Fouquet and J.M. Vanherpe
L.I.F.O., Facult´e des Sciences, B.P. 6759
0 Universit´e d’Orl´eans, 45067 Orl´eans Cedex 2, FR
1
0
2
Abstract

l
u In a graph G of maximum degree ∆ let γ denote the largest fraction
J of edges that can be ∆ edge-coloured. Albertson and Haas showed that
5 γ ≥ 1153 whenGiscubic[1]. Weshowherethatthisresultcanbeextended
1 to graphs with maximum degree 3 with the exception of a graph on 5
vertices. Moreover, thereareexactly twographswith maximumdegree3

] (onebeingobviously thePetersen graph)for which γ = 13. Thisextends
M 15
a result given in [14]. These results are obtained by giving structural
D properties of the so called δ−minimum edge colourings for graphs with
maximum degree 3.
.
s Keywords : Cubic graph; Edge-colouring;
c
Mathematics Subject Classification (2010) : 05C15.
[

4
v 1 Introduction
7
4
Throughoutthispaper,weshallbeconcernedwithconnectedgraphswithmax-
7
imum degree 3. We know by Vizing’s theorem [15] that these graphs can be
4
. edge-coloured with 4 colours. Let φ : E(G) → {α,β,γ,δ} be a proper edge-
9
colouring of G. It is often of interest to try to use one colour (say δ) as few as
0
possible. When an edge colouring is optimal, following this constraint, we shall
8
0 say that φ is δ−minimum. In [3] we gave without proof (in French, see [5] for
: a translation) results on δ−minimum edge-colourings of cubic graphs. Some
v
of them have been obtained later and independently by Steffen [13, 14]. Some
i
X other results which were not stated formally in [4] are needed here. The pur-
r pose ofSection2is to givethose resultsas structuralpropertiesofδ−minimum
a
edge-colourings as well as others which will be useful in Section 3.
An edge colouring of G using colours α,β,γ,δ is said to be δ-improper pro-
vided that adjacent edges having the same colours (if any) are colouredwith δ.
Itisclearthataproperedgecolouring(andhenceaδ−minimumedge-colouring)
ofGis aparticularδ-improperedgecolouring. Foraproperorδ-improperedge
colouring φ of G, it will be convenient to denote E (x) (x ∈ {α,β,γ,δ}) the
φ
set of edges coloured with x by φ. For x,y ∈ {α,β,γ,δ},x 6= y, φ(x,y) is the
partialsubgraphofGspannedbythese twocolours,thatisE (x)∪E (y)(this
φ φ
subgraph being a union of paths and even cycles where the colours x and y
alternate). Since any two δ−minimum edge-colourings of G have the same
number of edges coloured δ we shall denote by s(G) this number (the colour
number as defined in [13]).
1
As usual, for any undirected graph G, we denote by V(G) the set of its
vertices and by E(G) the set of its edges and we suppose that |V(G)| =n and
|E(G)|=m. AstrongmatchingC inagraphGisamatchingC suchthatthere
is no edgeofE(G) connecting anytwoedgesofC, or,equivalently,suchthatC
is the edge-set of the subgraph of G induced on the vertex-set V(C).
2 On δ−minimum edge-colouring
The graph G considered in the following series of Lemmas will have maximum
degree 3.
Lemma 1 [3,4,5]Any2-factor of Gcontains at least s(G)disjoint odd cycles.
Lemma 2 [3, 4, 5] Let φ be a δ−minimum edge-colouring of G. Any edge in
E (δ) is incident to α, β and γ. Moreover each such edge has one end of degree
φ
2 and the other of degree 3 or the two ends of degree 3.
Lemma 3 Let φ be a δ−improper colouring of G then there exists a proper
colouring of G φ′ such that E (δ)⊆E (δ)
φ′ φ
Proof Let φ be a δ-improper edge colouring of G. If φ is a proper colouring,
we are done. Hence, assume that uv and uw are colouredδ. If d(u)=2 we can
change the colour of uv to α,β or γ since v is incident to at most two colours
in this set.
If d(u) = 3 assume that the third edge uz incident to u is also coloured δ,
then we can change the colour of uv for the same reason as above.
If uz is coloured with α,β or γ, then v and w are incident to the two re-
maining colours of the set {α,β,γ} otherwise one of the edges uv, uw can be
recolouredwiththemissingcolour. W.l.o.g.,considerthatuz iscolouredαthen
v and w are incident to β and γ. Since u has degree 1 in φ(α,β) let P be the
path of φ(α,β) which ends on u. We can assume that v or w (say v) is not
the other end vertex of P. Exchanging α and β along P does not change the
colours incident to v. But now uz is coloured α and we can change the colour
of uv with β.
Ineachcase,wegethenceanewδ-improperedgecolouringφ withE (δ)(
1 φ1
E (δ). Repeating this process leads us to construct a proper edge colouring of
φ
G with E (δ)⊆E (δ) as claimed. (cid:3)
φ′ φ
Proposition 4 Let v ,v ...,v ∈V(G) such that G−{v ,v ,...v } is 3-edge
1 2 k 1 2 k
colourable. Then s(G)≤k.
Proof Letusconsidera3-edgecolouringofG−{v ,v ,...v }withα,β andγ
1 2 k
andletuscolourtheedgesincidenttov ,v ...,v withδ. Wegetaδ-improper
1 2 k
edge colouring φ of G. Lemma 3 gives a proper colouring of G φ′ such that
E (δ)⊆E (δ). Hence φ′ has at most k edges coloured with δ and s(G)≤k
φ′ φ
(cid:3)
Proposition4 above has been obtained by Steffen [13] for cubic graphs.
2
Lemma 5 Let φ be a δ−improper colouring of G then |E (δ)|≥s(G)
φ
Proof Applying Lemma 3, let φ′ be a proper edge colouring of G such that
E (δ)⊆E (δ). We clearly have |E (δ)|≥|E (δ)|≥s(G) (cid:3)
φ′ φ φ φ′
Lemma 6 [3, 4, 5] Let φ be a δ−minimum edge-colouring of G. For any edge
e=uv ∈E (δ)therearetwocolours xandy in{α,β,γ}suchthattheconnected
φ
component of φ(x,y) containing the two ends of e is an even path joining these
two ends.
Remark 7 An edge of E (δ) is in A when its ends can be connected by a
φ φ
path of φ(α,β), B by a path of φ(β,γ) and C by a path of φ(α,γ). It is
φ φ
clear that A , B and C are not necessarily pairwise disjoint since an edge of
φ φ φ
E (δ) with one end of degree 2 is containedin 2 such sets. Assume indeed that
φ
e=uv ∈E (δ)withd(u)=3andd(v)=2then,ifuisincidenttoαandβ and
φ
v isincidenttoγ wehaveanalternatingpathwhoseendsareuandv inφ(α,γ)
aswellasinφ(β,γ). Hence e isinA ∩B . Whene∈A wecanassociateto e
φ φ φ
the oddcycle C (e) obtainedby consideringthe path ofφ(α,β) together with
Aφ
e. We define in the same way C (e) and C (e) when e is in B or C . In
Bφ Cφ φ φ
the following lemma we consider an edge in A , an analogous result holds true
φ
whenever we consider edges in B or C as well.
φ φ
Lemma 8 [3, 4, 5] Let φ be a δ−minimum edge-colouring of G and let e be an
edge in A then for any edge e′ ∈C (e) there is a δ−minimum edge-colouring
φ Aφ
φ′ such that E (δ) = E (δ)−{e}∪{e′}, e′ ∈ A and C (e) = C (e′).
φ′ φ φ′ Aφ Aφ′
Moreover, each edge outside C (e) but incident with this cycle is coloured γ,
Aφ
φ and φ′ only differ on the edges of C (e).
Aφ
Foreachedge e∈E (δ)(where φ is a δ−minimumedge-colouring ofG) we
φ
can associate one or two odd cycles following the fact that e is in one or two
sets among A , B or C . Let C be the set of odd cycles associatedto edges in
φ φ φ
E (δ).
φ
Lemma 9 [3, 4, 5] For each cycle C ∈C, there are no two consecutive vertices
with degree two.
Lemma 10 [3,4,5]Lete ,e ∈E (δ)andletC ,C ∈C besuchthatC 6=C ,
1 2 φ 1 2 1 2
e ∈E(C ) and e ∈E(C ) then C and C are (vertex) disjoint.
1 1 2 2 1 2
By Lemma 8 any two cycles in C corresponding to edges in distinct sets
A , B or C are at distance at least 2. Assume that C = C (e ) and
φ φ φ 1 Aφ 1
C = C (e ) for some edges e and e in A . Can we say something about
2 Aφ 2 1 2 φ
the subgraph of G whose vertex set is V(C )∪V(C ) ? In general, we have
1 2
no answer to this problem. However, when G is cubic and any vertex of G
lies on some cycle of C (we shall say that C is spanning), we have a property
which will be useful later. Let us remark first that whenever C is spanning,
we can consider that G is edge-coloured in such a way that the edges of the
cycles of C are alternatively coloured with α and β (except one edge coloured
δ)andtheremainingperfectmatchingiscolouredwithγ. Forthisδ−minimum
edge-colouring of G we have B =∅ as well as C =∅.
φ φ
3
Lemma 11 Assume that G is cubic and C is spanning. Let e ,e ∈A and let
1 2 φ
C ,C ∈C such that C =C (e ) and C =C (e ). Then at least one of the
1 2 1 Aφ 1 2 Aφ 2
followings is true
(i) C and C are at distance at least 2
1 2
(ii) C and C are joined by at least 3 edges
1 2
(iii) C and C have at least two chords each
1 2
Proof Since e ,e ∈ A and C is spanning we have B = C = ∅. Let
1 2 φ φ φ
C = v v ...v and C = w w ...w . Assume that C and C are joined
1 0 1 2k1 2 0 1 2k2 1 2
by the edge v w . By Lemma 6, we can consider a δ−minimum edge-colouring
0 0
φ such that φ(v v ) = φ(w w ) = δ, φ(v v ) = φ(w w ) = β and φ(v v ) =
0 1 0 1 1 2 1 2 0 2k1
φ(w w ) = α. Moreover each edge of G (in particular v w ) incident with
0 2k2 0 0
these cycles is coloured γ. We can change the colour of v w in β. We obtain
0 0
thusanewδ−minimumedge-colouring φ′. Performingthatexchangeofcolours
on v w transforms the edges coloured δ v v and w w in two edges of C
0 0 0 1 0 1 φ′
lying on odd cycles C′ and C′ respectively. We get hence a new set C′ =
1 2
C−{C ,C }∪{C′,C′} of odd cycles associated to δ−coloured edges in φ′.
1 2 1 2
From Lemma 8 C′ (C′ respectively) is at distance at least 2 from any cycle
1 2
in C−{C ,C }. Hence V(C′)∪V(C′)⊆V(C )∪V(C ). It is an easy task to
1 2 1 2 1 2
check now that (ii) or (iii) above must be verified. (cid:3)
Lemma 12 [3, 4, 5] Let e =uv be an edge of E (δ) such that v has degree
1 1 φ 1
2 in G. Then v is the only vertex in N(u) of degree 2 and for any edge e =
1 2
u v ∈E (δ), {e ,e } induces a 2K .
2 2 φ 1 2 2
Lemma 13 [3, 4, 5] Let e and e be two edges of E (δ). If e and e are
1 2 φ 1 2
contained in two distinct sets of A ,B or C then {e ,e } induces a 2K
φ φ φ 1 2 2
otherwise e ,e are joined by at most one edge.
1 2
Lemma 14 [3, 4, 5] Let e ,e and e bethree distinct edges of E (δ) contained
1 2 3 φ
in the same set A ,B or C . Then {e ,e ,e } induces asubgraph with at most
φ φ φ 1 2 3
four edges.
3 Applications and problems
3.1 On a result by Payan
In [10] Payan showed that it is always possible to edge-colour a graph of max-
imum degree 3 with three maximal matchings (with respect to the inclusion)
and introduced henceforth a notion of strong-edge colouring where a strong
edge-colouringmeans that one colour is a strong matching while the remaining
colours are usual matchings. Payan conjectured that any d−regular graph has
d pairwise disjoint maximal matchings and showed that this conjecture holds
true for graphs with maximum degree 3.
ThefollowingresulthasbeenobtainedfirstbyPayan[10],buthistechnique
does not exhibit explicitly the odd cycles associated to the edges of the strong
matching and their properties.
4
Theorem 15 Let G be a graph with maximum degree at most 3. Then G has a
δ−minimumedge-colouring φwhere E (δ) is a strongmatching and, moreover,
φ
any edge in E (δ) has its two ends of degree 3 in G.
φ
Proof Letφbeaδ−minimumedge-colouring ofG. FromLemma13,anytwo
edges of E (δ) belonging to distinct sets from among A ,B and C induce a
φ φ φ φ
strong matching. Hence, we have to find a δ−minimum edge-colouring where
eachsetA ,B orC inducesastrongmatching(withthesupplementaryprop-
φ φ φ
erty that the end vertices of these edges have degree 3). That means that we
can work on eachset A ,B and C independently. Without loss of generality,
φ φ φ
we only consider A here.
φ
Assume that A = {e ,e ,...e } and A′ = {e ,...e } (1 ≤ i ≤ k−1) is
φ 1 2 k φ 1 i
a strong matching and each edge of A′ has its two ends with degree 3 in G.
φ
Consider the edge e and let C =C (φ)=(u ,u ...u ) be the odd cycle
i+1 ei+1 0 1 2p
associated to this edge (Lemma 6).
Let us mark any vertex v of degree 3 on C with a + whenever the edge of
colourγ incidenttothisvertexhasitsotherendwhichisavertexincidenttoan
edgeofA′ andletusmarkv with−otherwise. By Lemma9avertexofdegree
φ
2 on C has its two neighbours of degree 3 and by Lemma 12 these two vertices
are marked with a −. By Lemma 14 we cannot have two consecutive vertices
marked with a +. Hence, C must have two consecutive vertices of degree 3
marked with − whatever is the number of vertices of degree 2 on C.
Let u and u be two vertices of C of degree 3 marked with − (j being
j j+1
taken modulo 2p+1). We can transform the edge colouring φ by exchanging
colours on C uniquely, in such a way that the edge of colour δ of this cycle is
u u . Intheresultingedgecolouringφ wehaveA =A −{e }∪{u u }
j j+1 1 φ1 φ i+1 j j+1
andA′ =A′ ∪{u u }isastrongmatchingwhereeachedgehasitstwoends
φ1 φ j j+1
ofdegree3. Repeatingthisprocessweareleftwithanewδ−minimumcolouring
φ′ where A is a strong matching. (cid:3)
φ′
Corollary 16 Let G be a graph with maximum degree 3 then there are s(G)
vertices ofdegree3pairwise non-adjacent v ...v suchthatG−{v ...v }
1 s(G) 1 s(G)
is 3-colourable.
Proof Pickavertexoneachedgecolouredδ inaδ-minimumcolouringφofG
where E (δ) is a strong matching (Theorem 15). We get a subset S of vertices
φ
satisfying our corollary. (cid:3)
Steffen [13] obtained Corollary 16 for bridgeless cubic graphs.
3.2 Parsimonious edge colouring
Let χ′(G) be the classical chromatic index of G. For convenience let
c(G)=max{|E(H)|:H ⊆G ,χ′(H)=3}
c(G)
γ(G)=
|E(G)|
Staton [12] (and independently Locke [9]) showed that whenever G is a cubic
graph distinct from K then G contains a bipartite subgraph (and hence a 3-
4
edge colourable graph, by K¨onig’s theorem [8]) with at least 7 of the edges of
9
5
G. Bondy and Locke [2] obtained 4 when considering graphs with maximum
5
degree at most 3.
In[1]AlbertsonandHaasshowedthatwheneverGisacubicgraph,wehave
γ(G) ≥ 13 while for graphs with maximum degree 3 they obtained γ(G)≥ 26.
15 31
Our purpose here is to show that 13 is a lower bound for γ(G) when G has
15
maximum degree 3, with the exception of the graph G depicted in Figure 1
5
below.
Figure 1: G
5
Lemma 17 Let G be a graph with maximum degree 3 then γ(G)=1− s(G).
m
Proof Let φ be a δ−minimum edge-colouring of G. The restriction of φ
to E(G) − E (δ) is a proper 3-edge-colouring, hence c(G) ≥ m − s(G) and
φ
γ(G)≥1− s(G).
m
If H is a subgraph of G with χ(H)= 3, consider a proper 3-edge-colouring
φ : E(H) → {α,β,γ} and let ψ : E(G) → {α,β,γ,δ} be the continuation of
φ with ψ(e) = δ for e ∈ E(G)−E(H). By Lemma 3 there is a proper edge-
colouring ψ′ of G with E (δ) ⊆ E (δ) so that |E(H)| = |E(G)−E (δ)| ≤
ψ′ ψ ψ
|E(G)−E (δ)|≤m−s(G), c(G)≤m−s(G) and γ(G)≤1− s(G). (cid:3)
ψ′ m
In [11], Rizzi shows that for triangle free graphs of maximum degree 3,
γ(G) ≥ 1− 2 (where the odd girth of a graph G, denoted by g (G), is
3godd(G) odd
the minimum length of an odd cycle).
Theorem 18 Let G be a graph with maximum degree 3 then γ(G) ≥ 1 −
2 .
3godd(G)
Proof Letφbeaδ−minimumedge-colouring ofGandE (δ)={e ,e ...e }.
φ 1 2 s(G)
C being the set of odd cycles associated to edges in E (δ), we write C =
φ
{C ,C ...C } and suppose that for i = 1,2...s(G), e is an edge of C .
1 2 s(G) i i
We know by Lemma 10 that the cycles of C are vertex-disjoint.
Let us write C =C ∪C , where C denotes the set of odd cycles of C which
2 3 2
havea vertexof degree2,while C is for the setof cyclesin C whoseallvertices
3
have degree 3. Let k =|C |, obviously we have 0≤k≤s(G) and C ∩C =∅.
2 2 3
If C ∈ C , we may assume that e has a vertex of degree 2 (see Lemma 8)
i 2 i
and we can associate to e another odd cycle say C′ (Remark 7) whose edges
i i
distinct from e form an even path using at least godd(G) edges which are not
i 2
6
edges of C . Hence, C ∪C′ contains at least 3g (G) edges. Consequently
i i i 2 odd
there are at least 32 ×k×godd(G) edges in [ (Ci∪Ci′).
Ci∈C2
When C ∈C , C contains at least g (G) edges, moreover, each vertex of
i 3 i odd
C being of degree 3, there are s(G)−k ×g (G) additionnal edges which are
i 2 odd
incident to a vertex of [ Ci.
Ci∈C3
Since C ∩C =∅ and C′ ∩C =∅ (1≤i,j ≤s(G), i6=j), we have
i j i j
3 s(G)−k 3
m≥ g (G)×k+(s(G)−k)×g (G)+ ×g (G)= ×s(G)×g (G).
odd odd odd odd
2 2 2
Consequently γ(G)=1− s(G) ≥1− 2 .
m 3godd(G) (cid:3)
Lemma 19 [1] Let G be a graph with maximum degree 3. Assume that v ∈
V(G) is such that d(v)=1 then γ(G)>γ(G−v).
A triangle T = {a,b,c} is said to be reducible whenever its neighbours are
distinct. WhenT is a reducibletrianglein G(Ghavingmaximumdegree3)we
can obtain a new graph G′ with maximum degree 3 by shrinking this triangle
into a single vertex and joining this new vertex to the neighbours of T in G.
Lemma 20 [1] Let G be a graph with maximum degree 3. Assume that T =
{a,b,c} is a reducible triangle and let G′ be the graph obtained by reduction of
this triangle. Then γ(G)>γ(G′).
Theorem 21 Let G be a graph with maximum degree 3,V be the set of vertices
2
2
with degree2in Gand V those of degree3. If G6=G then γ(G)≥1− 15 .
3 5 1+2|V2|
3|V3|
Proof FromLemma19andLemma20wecanconsiderthatGhasonlyvertices
of degree 2 or 3 and that G contains no reducible triangle.
AssumethatwecanassociateasetP ofatleast5distinctverticesofV for
e 3
eachedgee∈E (δ)inaδ−minimumedge-colouring φofG. Assumemoreover
φ
that
∀e,e′ ∈E (δ) P ∩P =∅ (1)
φ e e′
Then
s(G) s(G) |V3|
γ(G)=1− =1− ≥1− 5
m 3|V |+|V | 3|V |+|V |
2 3 2 2 3 2
Hence
2
γ(G)≥1− 15
1+ 2|V2|
3|V3|
It remains to see how to construct the sets P satisfying Property (1). Let
e
C be the set of odd cycles associated to edges in E (δ) (see Lemma 10). Let
φ
e∈E (δ), assume that e is contained in a cycle C ∈C of length 3. By Lemma
φ
10, the edges incident to that triangle have the same colour in {α,β,γ}. This
triangle is hence reducible, impossible. We can thus consider that each cycle of
7
C has length at least 5. By Lemma 2 and Lemma 12, we know that whenever
such a cycle contains vertices of V , their distance on this cycle is at least 3.
2
Which means that every cycle C ∈ C contains at least 5 vertices in V as soon
3
as C has length at least 7 or C has length 5 but does not contain a vertex of
V . For each edge e ∈ E (δ) contained in such a cycle we associate P as any
2 φ e
set of 5 vertices of V contained in the cycle.
3
TheremayexistedgesinE (δ)containedina5-cycleofC havingexactlyone
φ
vertex in V . Let C =a a a a a be such a cycle and assume that a ∈V . By
2 1 2 3 4 5 1 2
Lemma 2 and Lemma 14, a is the only vertex of degree 2 and by exchanging
1
colours along this cycle, we can suppose that a a ∈ E (δ). Since a ∈ V ,
1 2 φ 1 2
e=a a is containedina secondcycle C′ ofC (see Remark7). IfC′ containsa
1 2
vertexx∈V distinct froma ,a ,a anda then wesetP ={a ,a ,a ,a ,x}.
3 2 3 4 5 e 2 3 4 5
Otherwise C′ =a a a a a and G is isomorphic to G , impossible.
1 2 4 3 5 5
The sets {P | e ∈ E (δ)} are pairwise disjoint since any two cycles of C
e φ
associated to distinct edges in E (δ) are disjoint. Hence property 1 holds and
φ
the proof is complete. (cid:3)
Albertson and Haas [1] proved that γ(G) ≥ 26 when G is a graph with
31
maximum degree 3 and Rizzi [11] obtained γ(G) ≥ 6. From Theorem 21 we
7
get immediately γ(G) ≥ 13, a better bound. Let us remark that we get also
15
γ(G)≥ 13 by Theorem 18 as soon as g ≥5.
15 odd
Lemma 22 Let G be a cubic graph which can be factored into s(G) cycles of
length 5 and without reducible triangle. Then every 2-factor of G contains s(G)
cycles of length 5.
Proof Since G has no reducible triangle, all cycles in a 2-factor have length
atleast4. Let C be any 2-factorofG. Letus denote n the number of cyclesof
4
length 4, n the number of cycles of length 5 and n the number of cycles on
5 6+
at least 6 vertices in C. We have 5n +6n ≤5s(G)−4n .
5 6+ 4
Whenn +n >0,ifn >0thenn +n <n +6n6+ ≤ 5s(G)−4n4 ≤s(G)
4 6+ 6+ 5 6+ 5 5 5
and if n =0, we have n >0 and n ≤ 5s(G)−4n4 <s(G). A contradiction in
6+ 4 5 5
both cases with Lemma 1. (cid:3)
Corollary 23 Let G be a graph with maximum degree 3 such that γ(G)= 13.
15
Then G is a cubic graph which can be factored into s(G) cycles of length 5.
Moreover every 2-factor of G has this property.
Proof Theoptimumforγ(G)inTheorem21isobtainedwhenevers(G)= |V3|
5
and |V |=0. That is, G is a cubic graph admitting a 2-factor of s(G) cycles of
2
length 5. Moreoverby Lemma 20 G has no reducible triangle, the result comes
from Lemma 22. (cid:3)
As pointed out by Albertson and Haas [1], the Petersen graph with γ(G)=
13 supplies an extremal example for cubic graphs. Steffen [14] proved that the
15
only cubic bridgeless graph with γ(G) = 13 is the Petersen graph. In fact,
15
we can extend this result to graphs with maximum degree 3 where bridges
are allowed(excluding the graphG ). Let P′ be the cubic graphon 10 vertices
5
8
obtainedfromtwocopiesofG (Figure1)byjoiningbyanedgethetwovertices
5
of degree 2.
Theorem 24 Let G be a connected graph with maximum degree 3 such that
γ(G)= 13. Then G is isomorphic to the Petersen graph or to P′.
15
Proof Let G be a graph with maximum degree 3 such that γ(G)= 13.
15
From Corollary 23, we can consider that G is cubic and G has a 2-factor of
cycles of length 5. Let C = {C ...C } be such a 2-factor ( C is spanning).
1 s(G)
Let φ be a δ−minimum edge-colouring of G induced by this 2−factor.
Without loss of generality consider two cycles in C, namely C and C ,
1 2
and let us denote C = v v v v v while C = u u u u u and assume that
1 1 2 3 4 5 2 1 2 3 4 5
v u ∈ G. From Lemma 11, C and C are joined by at least 3 edges or each
1 1 1 2
of them has two chords. If s(G) > 2 there is a cycle C ∈ C. Without loss of
3
generality, G being connected, we can suppose that C is joined to C by an
3 1
edge. Applying once more time Lemma 11, C and C have two chords or are
1 3
joined by at least 3 edges, contradiction with the constraints imposed by C
1
and C . Hence s(G) = 2 and G has 10 vertices and no 4-cycle, which leads to
2
a graph isomorphic to P′ or the Petersen graph as claimed. (cid:3)
We can construct cubic graphs with chromatic index 4 (snarks in the liter-
ature) which are cyclically 4- edge connected and having a 2-factor of C ’s.
5
Indeed, let G be a cubic cyclically 4-edge connected graph of order n and
cMopibees oafptherefePcettemrsaetnchgirnagpho.f FGo,rMeac=hP{x(iiyi=|i1=..1..n.).wn2e}.coLnestidPer1t.w..oPen2dgbeesan2t
i 2
distance 1 aparte1 and e2. Letus observethat P −{e1,e2} containsa 2-factor
i i i i i
of two C ’s (C1 and C2).
5 i i
We construct then a new cyclically 4-edge connected cubic graph H with
chromaticindex4byapplyingthewellknownoperationdot-product(seeFigure
2 for a description and [7] for a formal definition) on {e1,e2} and the edge x y
i i i i
(i=1...n). We remark that the vertices of G vanish in the operation and the
2
resultinggraphH hasa2factorofC ,namely{C1,C2,...C1,C2,...C1,C2}.
5 1 1 i i n n
2 2
We do not know example an of a cyclically 5-edge connected snark (except
the Petersen graph) with a 2-factor of induced cycles of length 5.
Problem 25 Is there any cyclically 5-edge connected snark distinct from the
Petersen graph with a 2-factor of C ’s ?
5
As a first step towards the resolution of this Problem we propose the fol-
lowing Theorem. Recall that a permutation graph is a cubic graphhaving some
2-factor with precisely 2 odd cycles.
Theorem 26 Let G be a cubic graph which can be factored into s(G) induced
odd cycles of length at least 5, then G is a permutation graph. Moreover, if G
has girth 5 then G is the Petersen graph.
Proof Let F be a 2-factor of s(G) cycles of length at least 5 in G, every
cycleofF beinganinducedoddcycleofG. We considertheδ−minimumedge-
colouring φ such that the edges of all cycles of F are alternatively coloured α
and β except for exactly one edge per cycle which is coloured with δ, all the
9
r a
x
s b
t c
y
u d
G1 G2
r a
s b
t c
u d
G .G
1 2
Figure 2: The dot product operation on graphs G , G .
1 2
10