188 research outputs found
On Colorings of Graph Powers
In this paper, some results concerning the colorings of graph powers are
presented. The notion of helical graphs is introduced. We show that such graphs
are hom-universal with respect to high odd-girth graphs whose st power
is bounded by a Kneser graph. Also, we consider the problem of existence of
homomorphism to odd cycles. We prove that such homomorphism to a -cycle
exists if and only if the chromatic number of the st power of
is less than or equal to 3, where is the 2-subdivision of . We also
consider Ne\v{s}et\v{r}il's Pentagon problem. This problem is about the
existence of high girth cubic graphs which are not homomorphic to the cycle of
size five. Several problems which are closely related to Ne\v{s}et\v{r}il's
problem are introduced and their relations are presented
Uniquely D-colourable digraphs with large girth
Let C and D be digraphs. A mapping is a C-colouring if for
every arc of D, either is an arc of C or , and the
preimage of every vertex of C induces an acyclic subdigraph in D. We say that D
is C-colourable if it admits a C-colouring and that D is uniquely C-colourable
if it is surjectively C-colourable and any two C-colourings of D differ by an
automorphism of C. We prove that if a digraph D is not C-colourable, then there
exist digraphs of arbitrarily large girth that are D-colourable but not
C-colourable. Moreover, for every digraph D that is uniquely D-colourable,
there exists a uniquely D-colourable digraph of arbitrarily large girth. In
particular, this implies that for every rational number , there are
uniquely circularly r-colourable digraphs with arbitrarily large girth.Comment: 21 pages, 0 figures To be published in Canadian Journal of
Mathematic
Between 2- and 3-colorability
We consider the question of the existence of homomorphisms between
and odd cycles when . We show that for any positive integer
, there exists such that if then
w.h.p. has a homomorphism from to so long as
its odd-girth is at least . On the other hand, we show that if
then w.h.p. there is no homomorphism from to . Note that in our
range of interest, w.h.p., implying that there is a
homomorphism from to
Circular edge-colorings of cubic graphs with girth six
We show that the circular chromatic index of a (sub)cubic graph with girth at
least six is at most 7/2.Comment: 13 pages, 6 figure
Graphs with bounded tree-width and large odd-girth are almost bipartite
We prove that for every and every , there exists such
that every graph with tree-width at most and odd-girth at least has
circular chromatic number at most
D-colorable digraphs with large girth
In 1959 Paul Erdos (Graph theory and probability, Canad. J. Math. 11 (1959), 34-38) famously proved, nonconstructively, that there exist graphs that have both arbitrarily large girth and arbitrarily large chromatic number. This result, along with its proof, has had a number of descendants (D. Bokal, G. Fijavz, M. Juvan, P.M. Kayll and B. Mohar, The circular chromatic number of a digraph, J. Graph Theory 46 (2004), 227-240; B. Bollobas and N. Sauer, Uniquely colourable graphs with large girth, Canad. J. Math. 28 (1976), 1340-1344; J. Nesetril and X. Zhu, On sparse graphs with given colorings and homomorphisms, J. Combin. Theory Ser. B 90 (2004), 161-172; X. Zhu, Uniquely H-colorable graphs with large girth, J. Graph Theory 23 (1996), 33-41) that have extended and generalized the result while strengthening the techniques used to achieve it. We follow the lead of Xuding Zhu (op. cit.) who proved that, for a suitable graph H, there exist graphs of arbitrarily large girth that are uniquely H-colorable. We establish an analogue of Zhu\u27s results in a digraph setting.
Let C and D be digraphs. A mapping f:V(D)&rarr V(C) is a C-coloring if for every arc uv of D, either f(u)f(v) is an arc of C or f(u)=f(v), and the preimage of every vertex of C induces an acyclic subdigraph in D. We say that D is C-colorable if it admits a C-coloring and that D is uniquely C-colorable if it is surjectively C-colorable and any two C-colorings of D differ by an automorphism of C. We prove that if D is a digraph that is not C-colorable, then there exist graphs of arbitrarily large girth that are D-colorable but not C-colorable. Moreover, for every digraph D that is uniquely D-colorable, there exists a uniquely D-colorable digraph of arbitrarily large girth
Circular choosability
International audienceWe study circular choosability, a notion recently introduced by Mohar and by Zhu. First, we provide a negative answer to a question of Zhu about circular cliques. We next prove that cch(G) = O(ch(G) + ln |V(G)|) for every graph G. We investigate a generalisation of circular choosability, the circular f-choosability, where f is a function of the degrees. We also consider the circular choice number of planar graphs. Mohar asked for the value of Ď„ := sup {cch(G) : G is planar}, and we prove that 68, thereby providing a negative answer to another question of Mohar. We also study the circular choice number of planar and outerplanar graphs with prescribed girth, and graphs with bounded density
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