Cubic bipartite cyclic 4-connected graphs without Hamiltonian circuits

A k-colouring of a graph G with colours 1 , 2 , … , k is canonical with respect to an ordering π = v 1 , v 2 , … , v n of the vertices of G if adjacent vertices are assigned different colours and, for 1 ≤ c ≤ k , whenever colour c is assigned to a vertex v i , each colour less than c has been assigned to a vertex that precedes v i in π . The canonical k-colouring graph of G with respect to π is the graph Can k π ( G ) with vertex set equal to the set of canonical k-colourings of G with respect to π , with two of these being adjacent if and only if they differ in the colour assigned to exactly one vertex. Connectivity and Hamiltonicity of canonical colouring graphs of bipartite and complete multipartite graphs is studied. It is shown that for complete multipartite graphs, and bipartite graphs there exists a vertex ordering π such that Can k π ( G ) is connected for large enough values of k. It is proved that a canonical colouring graph of a complete multipartite graph usually does not have a Hamilton cycle, and that there exists a vertex ordering π such that Can k π ( K m , n ) has a Hamilton path for all k ≥ 3 . The paper concludes with a detailed consideration of Can k π ( K 2 , 2 , … , 2 ) . For each k ≥ χ and all vertex orderings π , it is proved that Can k π ( K 2 , 2 , … , 2 ) is either disconnected or isomorphic to a particular tree.

Let $$\mathcal{B}$$B denote the class of all 3-connected cubic bipartite plane graphs. A conjecture of Barnette states that every graph in $$\mathcal{B}$$B has a Hamilton cycle. A cyclic sequence of big faces is a cyclic sequence of different faces, each bounded by at least six edges, such that two faces from the sequence are adjacent if and only if they are consecutive in the sequence. Suppose that $${F_1, F_2, F_3}$$F1,F2,F3 is a proper 3-coloring of the faces of $$G^*\in \mathcal{B}$$G??B. We prove that if every cyclic sequence of big faces of $$G^*$$G? has a face belonging to $$F_1$$F1 and a face belonging to $$F_2$$F2, then $$G^*$$G? has the following properties: $$H^{+-}$$H+-: If any two edges are chosen on the same face, then there is a Hamilton cycle through one and avoiding the other, $$H^{--}$$H--: If any two edges are chosen which are an even distance apart on the same face, then there is a Hamilton cycle that avoids both. Moreover, let $$X$$X and $$Y$$Y partition the set of big faces of $$G^*$$G? such that all such faces of $$F_1$$F1 are in $$X$$X and all such faces of $$F_2$$F2 are in $$Y$$Y. We prove that if every cyclic sequence of big faces has a face belonging to $$X$$X and a face belonging to $$Y$$Y, then $$G^*$$G? has a Hamilton cycle.

The dominating graph of a graph H has as its vertices all dominating sets of H, with an edge between two dominating sets if one can be obtained from the other by the addition or deletion of a single vertex of H. In this paper we prove that the dominating graph of any tree has a Hamilton path. We also show how a result about binary strings leads to a proof that the dominating graph of a cycle on n vertices has a Hamilton path if and only if n is not a multiple of 4.

A note on the middle levels problem

A short proof of the middle-levels theorem, Discrete Analysis 2018:8, 12 pp. Let $n$ be a positive integer, and define a bipartite graph where one vertex set consists of all subsets of $\{1,2,\dots,2n+1\}$ of size $n$, the other consists of all subsets of size $n+1$, and we join a set $A$ of size $n$ to a set $B$ of size $n+1$ if and only if $A\subset B$. The following innocent looking question was an open problem for a long time: does this graph contain a Hamilton cycle? That is, can we find a sequence of sets $A_1,B_1,\dots,A_N,B_N$ (where $N=\binom{2n+1}n$) such that each set of size $n$ or $n+1$ appears exactly once, and such that $A_1$ is contained in $B_1$, which contains $A_2$, which is contained in $B_2$, and so on all the way up to $B_N$, which contains $A_1$? For very small values of $n$, one can find Hamilton cycles, and one can even imagine that one is finding them in a potentially systematic way, but in the end all patterns seem to evaporate. One difficulty is that it can be shown that in any Hamilton cycle, each of the $2n+1$ possible edge directions must occur the same number of times. This means that if one tries to use a solution to the $n-1$ case to build a Hamilton cycle for the $n$ case, one will have to chop it up into exponentially many pieces. (This is in sharp contrast with the case of the entire $n$-dimensional cube, where one can easily prove that it contains a Hamilton cycle by taking two parallel Hamilton cycles in copies of an $(n-1)$-dimensional cube, removing an edge from the first and the corresponding edge from the second, and adding two linking edges to form a single Hamilton cycle.) Another approach that one might imagine attempting to use is a probabilistic one, perhaps creating most of a Hamilton cycle randomly and then making a few adjustments at the end in order to complete it. But this kind of argument also runs into problems: the constraints are not too severe to start with, but the more of the cycle one constructs, the stronger they become, and without a clever idea one gets stuck long before completing a cycle. In 2014 the problem, by then over 30 years old, was solved by Torsten Mutze [1], one of the authors of this paper, using a long and technical (and constructive) argument. This paper gives a much shorter and simpler argument that avoids nearly all the technicalities of the first argument and thereby makes Mutze's remarkable result truly accessible for the first time. Very roughly, the idea behind the proof is to begin by finding a certain special collection $\mathcal C$ of vertex-disjoint cycles that includes all vertices. If there were only one cycle in the collection, we would be done, but since that is not the case, we have to join them together. This is done by using an auxiliary collection of 6-cycles: if a 6-cycle intersects the union of two cycles $C_1,C_2$ in a particular way, then the symmetric difference of all three cycles is a single cycle that visits the same vertices as $C_1$ and $C_2$. The authors prove various lemmas about the cycles in $\mathcal C$ and the 6-cycles, which they then use to prove that this process can be continued until there is just one cycle left. (The reason 6-cycles are used, rather than 4-cycles as in the case of the complete cube, is brutally simple: it is easy to check that the middle-layers graph does not contain any 4-cycles.) The proof gives rise to an efficient algorithm for actually finding Hamilton cycles in the middle-layers graph. If you are curious to see what the cycles look like, the algorithm can be run online on [this web page](http://page.math.tu-berlin.de/~muetze/cos/middle.html). [1] Torsten Mutze, _Proof of the middle levels conjecture_, Proc. London Math. Soc. *112* (2016), pp. 677-713. Also available at [arXiv:1404.4442](https://arxiv.org/abs/1404.4442)

An edge-colored graph G is rainbow k-connected, if for every two vertices of G, there are k internally disjoint rainbow paths, i.e., if no two edges of each path are colored the same. The minimum number of colors needed for which there exists a rainbow k-connected coloring of G, $$rc_k(G)$$ , is the rainbow k-connection number of G. Let G and H be two connected graphs, where O is an orientation of G. Let $${\vec {e}}$$ be an oriented edge of H. The edge-comb product of G (under the orientation O) and H on $$\vec {e}$$ , $$G{}^o\rhd _{\vec {e}}H$$ , is a graph obtained by taking one copy of G and |E(G)| copies of H and identifying the i-th copy of H at the edge $${\vec {e}}$$ to the i-th edge of G, where the two edges have the same orientation. In this paper, we provide sharp lower and upper bounds for rainbow 2-connection numbers of edge-comb product of a cycle and a Hamiltonian graph. We also determine the rainbow 2-connection numbers of edge-comb product of a cycle with some graphs, i.e. complete graph, fan graph, cycle graph, and wheel graph.

A 1-factorisation of a graph G is a decomposition of G into edge-disjoint 1-factors (1-regular spanning subgraphs). A perfect 1-factorisation is a 1-factorisation in which the union of every pair of distinct 1-factors is a Hamilton cycle; a 1-factorisation is uniform if the graphs resulting from the union of every pair of distinct 1-factors are isomorphic. We initiate the study of perfect 1-factorisations and uniform 1-factorisations of circulant graphs, a well-studied family of Cayley graphs, by determining whether or not such 1-factorisations exist for even order circulant graphs of small degree. For a bipartite circulant graph, we observe that having order congruent to 2 (mod 4) is a necessary condition for the existence of a perfect 1-factorisation. We prove that for bipartite 3-regular circulant graphs this necessary condition is also sufficient. Furthermore, the bipartite 3-regular circulant graphs of order 2 (mod 4) are the only 3-regular circulant graphs of order greater than 6 which admit a perfect 1-factorisation. The 4-regular case is more complex; we provide constructions of perfect 1-factorisations for many families of bipartite 4-regular circulant graphs of order 2 (mod 4); however, we also prove that there is an infinite family of such graphs no member of which admits a perfect 1-factorisation. We conjecture that a connected non-bipartite 4-regular circulant graph of order at least 8 does not admit a perfect 1-factorisation; we prove this conjecture for three distinct infinite families of non-bipartite 4-regular circulant graphs and show by computer search that the conjecture holds for small orders (at most 30). Several of our results are achieved by studying perfect 1-factorisations of a particular family of 4-regular graphs, denoted Dh, k for integers h, knwhere h g 2 and k g 3. These graphs are shown to be isomorphic to Cayley graphs when k is even and yet they are not vertex-transitive when k is odd (and hng 3). For small values of h, we make use of the structure of Dh,k to determine when it admits a perfect 1-factorisation and then use specific isomorphism results to obtain corresponding results for 4-regular circulant graphs. In this thesis the connected 3-regular circulant graphs that admit a perfect 1-factorisation are characterised and although the connected 4-regular circulant graphs that admit a perfect 1-factorisation have not yet been fully characterised, we provide many interesting theoretical results that establish the framework for such a characterisation.n

We consider the zeta function associated with the time evolution matrix of a bipartite walk on a bipartite graph and derive a formula for its absolute zeta function. Furthermore, we give formulas for the absolute zeta functions of the zeta functions of the time evolution matrices of bipartite walks on the complete bipartite graph and an even cycle graph. As an application, we obtain formulas for the absolute zeta functions of the zeta functions of the Grover matrices of the complete bipartite graph and an even cycle graph.

We study the problem of the location of real zeros of chromatic polynomials for some families of graphs. In particular, a problem presented by Thomassen (see [On the number of hamiltonian cycles in bipartite graphs, Combin. Probab. Comput. 5 (1996) 437-442.]) is discussed and a result for hamiltonian graphs is presented. An open problem is stated for 2-connected graphs with a hamiltonian path.

The Implementation Of Bipartite Graph To Minimize Crossing Number Problem Of Crossroads In Manado. Supervised by BENNY PINONTAN as main supervisor and CHRIESTIE E. J. C. MONTOLALU as co-supervisor. In general, the crossroads are the meeting points of two-way roads from four different places. This causes cross direction at that point. There are various methods that can be used to minimize the crossing number problem crossroad, for example graph theory. The ability of graph theory can later help describe crossroads in Manado into graph, with nodes and lines. In this case, the crossing number problems will solve by bipartite graph. Bipartite graph is a graph that does not have an odd cycle, and can be partitioned into two parts of a set of vertices. Based on results of this research, the appropriate form of the bipartite graph is and in two different form. First, with an isolated vertex, and second, without isolated vertex. In the case of crossroads, Bipartite graph turns out to be one method that is very suitable and helps determine the crossing number and its solution quickly.

We give a necessary and sufficient condition for a cubic graph to be Hamiltonian by analyzing Eulerian tours in certain spanning subgraphs of the quartic graph associated with the cubic graph by 1 -factor contraction. This correspondence is most useful in the case when it induces a blue and red 2 -factorization of the associated quartic graph. We use this condition to characterize the Hamiltonian I -graphs, a further generalization of generalized Petersen graphs. The characterization of Hamiltonian I -graphs follows from the fact that one can choose a 1 -factor in any I -graph in such a way that the corresponding associated quartic graph is a graph bundle having a cycle graph as base graph and a fiber and the fundamental factorization of graph bundles playing the role of blue and red factorization. The techniques that we develop allow us to represent Cayley multigraphs of degree 4 , that are associated to abelian groups, as graph bundles. Moreover, we can find a family of connected cubic (multi)graphs that contains the family of connected I -graphs as a subfamily.

Variations sur le thème [formula omitted

A decomposition of a graph is a set of subgraphs whose edges partition those of $G$. The 3-decomposition conjecture posed by Hoffmann-Ostenhof in 2011 states that every connected cubic graph can be decomposed into a spanning tree, a 2-regular subgraph, and a matching. It has been settled for special classes of graphs, one of the first results being for Hamiltonian graphs. In the past two years several new results have been obtained, adding the classes of plane, claw-free, and 3-connected tree-width 3 graphs to the list.
 In this paper, we regard a natural extension of Hamiltonian graphs: removing a Hamiltonian cycle from a cubic graph leaves a perfect matching. Conversely, removing a perfect matching $M$ from a cubic graph $G$ leaves a disjoint union of cycles. Contracting these cycles yields a new graph $G_M$. The graph $G$ is star-like if $G_M$ is a star for some perfect matching $M$, making Hamiltonian graphs star-like. We extend the technique used to prove that Hamiltonian graphs satisfy the 3-decomposition conjecture to show that 3-connected star-like graphs satisfy it as well.

In a graph G a matching is a set of edges in which no two edges have a common endpoint. An induced matching is a matching in which no two edges are linked by an edge of G. The maximum induced matching (abbreviated MIM) problem is to find the maximum size of an induced matching for a given graph G. This problem is known to be NP-hard even on bipartite graphs or on planar graphs. We present a polynomial time algorithm which given a graph G either finds a maximum induced matching in G, or claims that the size of a maximum induced matching in G is strictly less than the size of a maximum matching in G. We show that the MIM problem is NP-hard on line-graphs, claw-free graphs, chair-free graphs, Hamiltonian graphs and r-regular graphs for r \geq 5. On the other hand, we present polynomial time algorithms for the MIM problem on (P5,Dm)-free graphs, on (bull, chair)-free graphs and on line-graphs of Hamiltonian graphs.

A 2-factorization {F1, F2,…,Fd} of a 2d-regular graph G such that each [Formula: see text] and the remaining Fi's are all Hamilton cycles is called Hamilton cycle rich 2-factorization of G, where Gi's are the given non-isomorphic 2-factors of G. In this paper, we prove that there exists a 2-factorization {F1, F2,…,Fn} of K2n,2n such that F1 ≅ G1, F2 ≅ G2 and the remaining Fi's are Hamilton cycles of K2n,2n, where G1 and G2 are the given two non-isomorphic 2-factors of K2n,2n. In fact our result together with the earlier results settles the existence of Hamilton cycle rich 2-factorizations of K(m, p), the complete p-partite graph with m vertices in each partite set, except when (m, p) = (2n + 1, 2), in the case that two of the 2-factors are isomorphic to the given two non-isomorphic 2-factors and the remaining are Hamilton cycles.