Abstract
A class \(\mathcal{G}\) of simple graphs is said to be girth-closed (odd-girth-closed) if for any positive integer g there exists a graph \(\mathrm {G} \in \mathcal{G}\) such that the girth (odd-girth) of \(\mathrm {G}\) is \(\ge g\). A girth-closed (odd-girth-closed) class \(\mathcal{G}\) of graphs is said to be pentagonal (odd-pentagonal) if there exists a positive integer \(g^*\) depending on \(\mathcal{G}\) such that any graph \(\mathrm {G} \in \mathcal{G}\) whose girth (odd-girth) is greater than \(g^*\) admits a homomorphism to the five cycle (i.e. is \(\mathrm {C}_{_{5}}\)-colourable). Although, the question “Is the class of simple 3-regular graphs pentagonal?” proposed by Nešetřil (Taiwan J Math 3:381–423, 1999) is still a central open problem, Gebleh (Theorems and computations in circular colourings of graphs, 2007) has shown that there exists an odd-girth-closed subclass of simple 3-regular graphs which is not odd-pentagonal. In this article, motivated by the conjecture that the class of generalized Petersen graphs is odd-pentagonal, we show that finding the odd girth of generalized Petersen graphs can be transformed to an integer programming problem, and using the combinatorial and number theoretic properties of this problem, we explicitly compute the odd girth of such graphs, showing that the class is odd-girth-closed. Also, we obtain upper and lower bounds for the circular chromatic number of these graphs, and as a consequence, we show that the subclass containing generalized Petersen graphs \(\mathrm {Pet}(n,k)\) for which either k is even, n is odd and \(n\mathop {\equiv }\limits ^{k-1}\pm 2\) or both n and k are odd and \(n\ge 5k\) is odd-pentagonal. This in particular shows the existence of nontrivial odd-pentagonal subclasses of 3-regular simple graphs.
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References
Boben M, Pisanski T, Žitnic A (2005) I-graphs and the corresponding configurations. J Comb Des 13:406–424
Borodin OV, Hartke SG, Ivanova AO, Kostochka AV, West DB (2008) Circular \((5,2)\)-coloring of sparse graphs. Sib Élektron Mat Izv 5:417–426
Borodin OV, Kim SJ, Kostochka AV, West DB (2004) Homomorphisms from sparse graphs with large girth. J Comb Theory Ser B 90(1):147–159
Catlin PA (1988) Graph homomorphisms into the five-cycle. J Comb Theory Ser B 45:199–211
Coxeter HSM (1950) Self-dual configurations and regular graphs. Bull Amer Math Soc 56:413–455
Daneshgar A, Hajiabolhassan H (2008) Density and power graphs in graph homomorphism problem. Discret Math 308:4027–4030
Daneshgar A, Hejrati M, Madani M (2016) On cylindrical graph construction and its applications. Electron J Comb 23:P1.29
Fox J, Gera R, Stǎnicǎ P (2012) The independence number for the generalized Petersen graphs. Ars Comb 103:439–451
Frucht R, Graver JE, Watkins M (1971) The groups of the generalized Petersen graphs. Proc Camb Philos Soc 70:211–218
Galluccioa A, Goddyn LA, Hell P (2001) High-girth graphs avoiding a minor are nearly bipartite. J Comb Theory Ser B 83:1–14
Ghebleh M (2007) Theorems and computations in circular colourings of graphs, PhD. Thesis, Simon Fraser University
Hajiabolhassan H (2009) On colorings of graph powers. Discret Math 309:4299–4305
Hajiabolhassan H, Taherkhani A (2010) Graph powers and graph homomorphisms. Electron J Comb 17:R17
Hajiabolhassan H, Taherkhani A (2014) On the circular chromatic number of graph powers. J Graph Theory 75:48–58
Hatami H (2005) Random cubic graphs are not homomorphic to the cycle of size \(7\). J Comb Theory Ser B 93:319–325
Hell P, Nešetřil J (2004) Graph and homomorphisms., Oxford lecture series in mathematics and its applications, Oxford University Press, Oxford
Horvat B, Pisanski T, Žitnic A (2012) Isomorphism checking of I-graphs. Graphs Comb 28:823–830
Kostochka A, Nešetřil J, Smolíková P (2001) Colorings and homomorphisms of degenerate and bounded degree graphs. Discret Math 233:257–276
Nedela R, Škoviera M (1995) Which generalized Petersen graphs are Cayley graphs? J Graph Theory 19:1–11
Nešetřil J (1999) Aspects of structural combinatorics (graph homomorphisms and their use). Taiwan J Math 3:381–423
Pan Z, Zhu X (2002) The circular chromatic number of series-parallel graphs of large odd girth. Discret Math 245:235–246
Steimle A, Staton W (2009) The isomorphism classes of the generalized Petersen graphs. Discret Math 309:231–237
Vince A (1988) Star chromatic number. J Graph Theory 12:551–559
Wanless IM, Wormald NC (2001) Regular graphs with no homomorphisms onto cycles. J Comb Theory Ser B 82:155–160
Watkins ME (1969) A theorem on Tait colorings with an application to the generalized Petersen graphs. J Comb Theory 6:152–164
Yang C (2005) Perfectness of the complements of circular complete graphs, Master Thesis, National Sun Yat-sen University
Zhu X (2001) Circular chromatic number: a survey. Discret Math 229:371–410
Zhu X (2006) Recent developments in circular colouring of graphs. Topics in discrete mathematics of Algorithms Combinations. Springer, Berlin, pp 497–550
Žitnik A, Horvat B, Pisanski T (2012) All generalized Petersen graphs are unit-distance graphs. J Korean Math Soc 3:475–491
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The authors wish to express their sincere thanks to Ali Taherkhani and an anonymous referee for their constructive comments for improvement.
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A primary version of this article has already been posted on http://arxiv.org/abs/1501.06551.
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Daneshgar, A., Madani, M. On the odd girth and the circular chromatic number of generalized Petersen graphs. J Comb Optim 33, 897–923 (2017). https://doi.org/10.1007/s10878-016-0013-0
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DOI: https://doi.org/10.1007/s10878-016-0013-0