Abstract
We discuss the existence of homomorphisms to oriented cycles and give, for a special class of cyclesC, a characterization of those digraphs that admit, a homomorphism toC. Our result can be used to prove the multiplicativity of a certain class of oriented cycles, (and thus complete the characterization of multiplicative oriented cycles), as well as to prove the membership of the corresponding decision problem in the classNPϒcoNP. We also mention a conjecture on the existence of homomorphisms to arbitrary oriented cycles.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Bang-Jensen, andP. Hell: On the effect of two cycles on the complexity of colouring,Discrete Applied Math. 26 (1990), 1–23.
J. Bang-Jensen, P. Hell, andG. MacGillivray: The complexity of colouring by semicomplete digraphs,SIAM J. on Discrete Math. 1 (1988), 281–298.
J. Bang-Jensen, P. Hell, andG. MacGillivray.: On the complexity of colouring by superdigraphs of bipartite graphs.Discrete Math. 109 (1992), 27–44.
J. Bang-Jensen, P. Hell, andG. MacGillivray: Hereditarily hard colouring problems, submitted toDiscrete Math.
S. Burr, P. Erdős, andL. Lovász: On graphs of Ramsey type,Ars Comb. 1 (1976), 167–190.
D. Duffus, B. Sands, andR. Woodrow: On the chromatic number of the product of graphs,J. Graph Theory 9 (1985), 487–495.
H. El-Zahar, andN. Sauer: The chromatic number of the product of two 4-chromatic graphs is 4,Combinatorica,5 (1985), 121–126.
W. Gutjahr, E. Welzl, andG. Woeginger Polynomial graph colourings,Discrete Applied Math. 35 (1992), 29–46.
R. Häggkvist, P. Hell, D. J. Miller, andV. Neumann-Lara: On multiplicative graphs and the product conjecture,Combinatorica 8, (1988), 71–81.
R. Häggkvist, andP. Hell: OnA-mote universal graphs,European J. of Combinatorics 14 (1993), 23–27.
F. Harary:Graph Theory, Addison Wesley, 1969.
S. Hedetniemi: Homomorphisms and graph automata, University of Michigan Technical Report 03105-44-T, 1966.
P. Hell: An introduction to the category of graphs.,Annals of the N. Y. Acad. Sc. 328 (1979) 120–136.
P. Hell, andJ. Nešetřil: Homomorphisms of graphs and their orientations.,Monatshefte für Math. 85 (1978), 39–48.
P. Hell, andJ. Nešetřil: On the complexity ofH-colouring,J. Combin. Theory B 48 (1990), 92–100.
P. Hell, H. Zhou, andX. Zhu: Multiplicativity of oriented cycles,Combin. Theory B, in print.
P. Hell, andX. Zhu: Homomorphisms to oriented paths,Discrete Math, in print.
P. Hell, andX. Zhu: The existence of homomorphisms to unbalanced, cycles, submitted.
P. Komárek: Some new good characterizations of directed graphs,Časopis Pěst. Mat. 51 (1984), 348–354.
G. MacGillivray: On the complexity of colourings by vertex-transitive and arctransitive digraphs,SIAM J. Discrete Math. 4 (1991), 397–408.
H. A. Maurer, J. H. Sudborough, andE. Welzl: On the complexity of the general colouring problem,Inform. and Control 51 (1981), 123–145.
D. J. Miller: The categorical product of graphs,Canad. J. Math. 20 (1968), 1511–1521.
J. Nešetřil, andA. Pultr: On classes of relations and graphs determined by subobjects and factorobjects,Discrete Math. 22 (1978), 287–300.
N. Sauer, andX. Zhu: An approach to Hedetniemi's, conjecture.,J. Graph Theory, to appear.
E. Welzl: Symmetric graphs and interpretations,J. Combin. Th. (B) 37 (1984), 235–244.
H. Zhou: Homomorphism properties of graph products, Ph.D. thesis, Simon Fraser University, 1988.
X. Zhu: Multiplicative structures, Ph.D. thesis, The University of Calgary, 1990.
X. Zhu: A polynomial algorithm for homomorphisms to oriented cycles, manuscript, 1991.