Acyclic edge coloring of planar graphs without cycles of?specific lengths

A proper edge coloring of a graph G is called acyclic edge coloring if there are no bicolored cycles in G. Let ??(G) denote the maximum degree of G. In this paper, we prove that every planar graph with ??(G)??10 and without cycles of lengths 4 to 11 is acyclic (??(G)+1)-edge colorable, and every planar graph with ??(G)??11 and without cycles of lengths 4 to 9 is acyclic (??(G)+1)-edge colorable.     

Counterexamples to Grötzsch-Sachs-Koester's conjecture

Let G be a 4-regular planar graph and suppose that G has a cycle decomposition S (i.e., each edge of G is in exactly on cycle of the decomposition) with every pair of adjacent edges on a face always in different cycles of S. Such a graph G arises as a superposition of simple closed curves in the plane with tangencies disallowed. Grötzsch-Sachs-Koester's conjecture states that if the cycles of G can be partitioned into four classes, such that two cycles in the same classes are disjoint, G is vertex 3-colorable. In this note, the conjecture is disproved.     

Arbitrary decompositions into open and closed trails

The problem of arbitrary decomposition of a graph G into closed trails i.e. a decomposition into closed trails of prescribed lengths summing up to the size of the graph G was first considered in the case of the complete graph G=K_n (for odd n) in connection with vertex-distinguishing coloring of the union of cycles.Next, the same problem was investigated for other families of graphs.In this paper we consider a more general problem: arbitrary decomposition of a graph into open and closed trails. Our results are based on and generalize known results on decomposition of a graph into closed trails. Our results also generalize some results concerning decomposition of a graph into open trails. We here emphasize that the known results on the closed case are basic ingredients for the proof of the open and closed case.     

The structure of graphs with given lengths of cycles

The cycle spectrum of a given graph G is the lengths of cycles in G. In this paper, we introduce the following problem: determining the maximum number of edges of an n-vertex graph with given cycle spectrum. In particular, we determine the maximum number of edges of an n-vertex graph without containing cycles of lengths three and at least k. This can be viewed as an extension of a well-known result of Erd?s and Gallai concerning the maximum number of edges of an n-vertex graph without containing cycles of lengths at least k. We also determine the maximum number of edges of an n-vertex graph whose cycle spectrum is a subset of two positive integers.     

Subgraphs as circuits and bases of matroids

As is well known, the cycles of any given graph G may be regarded as the circuits of a matroid defined on the edge set of G. The question of whether other families of connected graphs exist such that, given any graph G, the subgraphs of G isomorphic to some member of the family may be regarded as the circuits of a matroid defined on the edge set of G led us, in two other papers, to the proof of some results concerning properties of the cycles when regarded as circuits of such matroids. Here we prove that the wheels share many of these properties with the cycles. Moreover, properties of subgraphs which may be regarded as bases of such matroids are also investigated.     

Partitioning 2-edge-colored graphs by monochromatic paths and cycles

We present results on partitioning the vertices of 2-edge-colored graphs into monochromatic paths and cycles. We prove asymptotically the two-color case of a conjecture of Sárközy: the vertex set of every 2-edge-colored graph can be partitioned into at most 2α(G) monochromatic cycles, where α(G) denotes the independence number of G. Another direction, emerged recently from a conjecture of Schelp, is to consider colorings of graphs with given minimum degree. We prove that apart from o(|V (G)|) vertices, the vertex set of any 2-edge-colored graph G with minimum degree at least \(\tfrac{{(1 + \varepsilon )3|V(G)|}} {4}\) can be covered by the vertices of two vertex disjoint monochromatic cycles of distinct colors. Finally, under the assumption that \(\bar G\) does not contain a fixed bipartite graph H, we show that in every 2-edge-coloring of G, |V (G)| ? c(H) vertices can be covered by two vertex disjoint paths of different colors, where c(H) is a constant depending only on H. In particular, we prove that c(C ₄)=1, which is best possible.     

A note on the acyclic 3-choosability of some planar graphs

An acyclic coloring of a graph G is a coloring of its vertices such that: (i) no two adjacent vertices in G receive the same color and (ii) no bicolored cycles exist in G. A list assignment of G is a function L that assigns to each vertex v∈V(G) a list L(v) of available colors. Let G be a graph and L be a list assignment of G. The graph G is acyclically L-list colorable if there exists an acyclic coloring ? of G such that ?(v)∈L(v) for all v∈V(G). If G is acyclically L-list colorable for any list assignment L with |L(v)|≥k for all v∈V(G), then G is said to be acyclically k-choosable. Borodin et al. proved that every planar graph with girth at least 7 is acyclically 3-choosable (Borodin et al., submitted for publication [4]). More recently, Borodin and Ivanova showed that every planar graph without cycles of length 4 to 11 is acyclically 3-choosable (Borodin and Ivanova, submitted for publication [7]). In this note, we connect these two results by a sequence of intermediate sufficient conditions that involve the minimum distance between 3-cycles: we prove that every planar graph with neither cycles of lengths 4 to 7 (resp. to 8, to 9, to 10) nor triangles at distance less than 7 (resp. 5, 3, 2) is acyclically 3-choosable.     

Hamilton decompositions of graphs with primitive complements

A k-factor of a graph is a k-regular spanning subgraph. A Hamilton cycle is a connected 2-factor. A graph G is said to be primitive if it contains no k-factor with 1≤k<Δ(G). A Hamilton decomposition of a graph G is a partition of the edges of G into sets, each of which induces a Hamilton cycle. In this paper, by using the amalgamation technique, we find necessary and sufficient conditions for the existence of a 2x-regular graph G on n vertices which:

1.

has a Hamilton decomposition, and

2.

has a complement in K_n that is primitive.

This extends the conditions studied by Hoffman, Rodger, and Rosa [D.G. Hoffman, C.A. Rodger, A. Rosa, Maximal sets of 2-factors and Hamiltonian cycles, J. Combin. Theory Ser. B 57 (1) (1993) 69-76] who considered maximal sets of Hamilton cycles and 2-factors. It also sheds light on construction approaches to the Hamilton-Waterloo problem.     

4-chromatic edge critical Grötzsch-Sachs graphs

Let G be a 4-regular plane graph and suppose that G has a cycle decomposition S (i.e., each edge of G is in exactly one cycle of the decomposition) with every pair of adjacent edges on a face always in different cycles of S. Such a graph G arises as a superposition of simple closed curves in the plane with tangencies disallowed. Two 4-chromatic edge critical graphs of order 48 generated by four curves are presented.     

A Min–Max Property of Chordal Bipartite Graphs with Applications

We show that if G is a bipartite graph with no induced cycles on exactly 6 vertices, then the minimum number of chain subgraphs of G needed to cover E(G) equals the chromatic number of the complement of the square of line graph of G. Using this, we establish that for a chordal bipartite graph G, the minimum number of chain subgraphs of G needed to cover E(G) equals the size of a largest induced matching in G, and also that a minimum chain subgraph cover can be computed in polynomial time. The problems of computing a minimum chain cover and a largest induced matching are NP-hard for general bipartite graphs. Finally, we show that our results can be used to efficiently compute a minimum chain subgraph cover when the input is an interval bigraph.     

An upper bound for the competition numbers of graphs

A hole of a graph is an induced cycle of length at least 4. Kim (2005) [2] conjectured that the competition number k(G) is bounded by h(G)+1 for any graph G, where h(G) is the number of holes of G. In Lee et al. [3], it is proved that the conjecture is true for a graph whose holes are mutually edge-disjoint. In Li et al. (2009) [4], it is proved that the conjecture is true for a graph, all of whose holes are independent. In this paper, we prove that Kim’s conjecture is true for a graph G satisfying the following condition: for each hole C of G, there exists an edge which is contained only in C among all induced cycles of G.     

Tutte’s 5-flow conjecture for highly cyclically connected cubic graphs

In 1954, Tutte conjectured that every bridgeless graph has a nowhere-zero 5-flow. Let ω(G) be the minimum number of odd cycles in a 2-factor of a bridgeless cubic graph G. Tutte’s conjecture is equivalent to its restriction to cubic graphs with ω≥2. We show that if a cubic graph G has no edge cut with fewer than edges that separates two odd cycles of a minimum 2-factor of G, then G has a nowhere-zero 5-flow. This implies that if a cubic graph G is cyclically n-edge connected and , then G has a nowhere-zero 5-flow.     

The spectral radius of graphs without paths and cycles of specified length

Let G be a graph with n vertices and μ(G) be the largest eigenvalue of the adjacency matrix of G. We study how large μ(G) can be when G does not contain cycles and paths of specified order. In particular, we determine the maximum spectral radius of graphs without paths of given length, and give tight bounds on the spectral radius of graphs without given even cycles. We also raise a number of open problems.     

The 1-factorization of some line-graphs

It is proved that if G is a simple graph with an even number of edges and such that its edge-set can be partitioned into Hamiltonian cycles, its line-graph is 1-factorable.     

Colouring Clique-Hypergraphs of Circulant Graphs

A clique-colouring of a graph G is a colouring of the vertices of G so that no maximal clique of size at least two is monochromatic. The clique-hypergraph, ${\mathcal{H}(G)}$ , of a graph G has V(G) as its set of vertices and the maximal cliques of G as its hyperedges. A vertex-colouring of ${\mathcal{H}(G)}$ is a clique-colouring of G. Determining the clique-chromatic number, the least number of colours for which a graph G admits a clique-colouring, is known to be NP-hard. In this work, we establish that the clique-chromatic number of powers of cycles is equal to two, except for odd cycles of size at least five, that need three colours. For odd-seq circulant graphs, we show that their clique-chromatic number is at most four, and determine the cases when it is equal to two. Similar bounds for the chromatic number of these graphs are also obtained.     

Packing edge-disjoint cycles in graphs and the cyclomatic number

For a graph G let μ(G) denote the cyclomatic number and let ν(G) denote the maximum number of edge-disjoint cycles of G.We prove that for every k≥0 there is a finite set P(k) such that every 2-connected graph G for which μ(G)−ν(G)=k arises by applying a simple extension rule to a graph in P(k). Furthermore, we determine P(k) for k≤2 exactly.     

On 3-choosable planar graphs of girth at least 4

Let G be a plane graph of girth at least 4. Two cycles of G are intersecting if they have at least one vertex in common. In this paper, we show that if a plane graph G has neither intersecting 4-cycles nor a 5-cycle intersecting with any 4-cycle, then G is 3-choosable, which extends one of Thomassen’s results [C. Thomassen, 3-list-coloring planar graphs of girth 5, J. Combin. Theory Ser. B 64 (1995) 101-107].     

Some remarks on the geodetic number of a graph

A set of vertices D of a graph G is geodetic if every vertex of G lies on a shortest path between two not necessarily distinct vertices in D. The geodetic number of G is the minimum cardinality of a geodetic set of G.We prove that it is NP-complete to decide for a given chordal or chordal bipartite graph G and a given integer k whether G has a geodetic set of cardinality at most k. Furthermore, we prove an upper bound on the geodetic number of graphs without short cycles and study the geodetic number of cographs, split graphs, and unit interval graphs.     

On cyclic edge-connectivity of transitive graphs

A cyclic edge-cut of a graph G is an edge set, the removal of which separates two cycles. If G has a cyclic edge-cut, then it is said to be cyclically separable. For a cyclically separable graph G, the cyclic edge-connectivity cλ(G) is the cardinality of a minimum cyclic edge-cut of G. In this paper, we first prove that for any cyclically separable graph G, , where ω(X) is the number of edges with one end in X and the other end in V(G)?X. A cyclically separable graph G with cλ(G)=ζ(G) is said to be cyclically optimal. The main results in this paper are: any connected k-regular vertex-transitive graph with k≥4 and girth at least 5 is cyclically optimal; any connected edge-transitive graph with minimum degree at least 4 and order at least 6 is cyclically optimal.