Even subgraphs of bridgeless graphs and 2-factors of line graphs

By Petersen's theorem, a bridgeless cubic multigraph has a 2-factor. Fleischner generalised this result to bridgeless multigraphs of minimum degree at least three by showing that every such multigraph has a spanning even subgraph. Our main result is that every bridgeless simple graph with minimum degree at least three has a spanning even subgraph in which every component has at least four vertices. We deduce that if G is a simple bridgeless graph with n vertices and minimum degree at least three, then its line graph has a 2-factor with at most max{1,(3n-4)/10} components. This upper bound is best possible.     

Characterization of reducible hexagons and fast decomposition of elementary benzenoid graphs

A benzenoid graph is a finite connected plane graph with no cut vertices in which every interior region is bounded by a regular hexagon of a side length one. A benzenoid graph G is elementary if every edge belongs to a 1-factor of G. A hexagon h of an elementary benzenoid graph is reducible, if the removal of boundary edges and vertices of h results in an elementary benzenoid graph. We characterize the reducible hexagons of an elementary benzenoid graph. The characterization is the basis for an algorithm which finds the sequence of reducible hexagons that decompose a graph of this class in O(n²) time. Moreover, we present an algorithm which decomposes an elementary benzenoid graph with at most one pericondensed component in linear time.     

The structure of even factors in claw-free graphs

Recently, Jackson and Yoshimoto proved that every bridgeless simple graph G with δ(G)≥3 has an even factor in which every component has order at least four, which strengthens a classical result of Petersen. In this paper, we give a strengthening of the above result and show that the above graphs have an even factor in which every component has order at least four that does not contain any given edge. We also extend the above result to the graphs with minimum degree at least three such that all bridges lie in a common path and to the bridgeless graphs that have at most two vertices of degree two respectively. Finally we use this extended result to show that every simple claw-free graph G of order n with δ(G)≥3 has an even factor with at most components. The upper bound is best possible.     

The upper bound of the number of cycles in a 2‐factor of a line graph

Let G be a simple graph with order n and minimum degree at least two. In this paper, we prove that if every odd branch‐bond in G has an edge‐branch, then its line graph has a 2‐factor with at most components. For a simple graph with minimum degree at least three also, the same conclusion holds. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 72–82, 2007     

On graphs whose square have strong hamiltonian properties

The squareG² of a graph G is the graph having the same vertex set as G and two vertices are adjacent if and only if they are at distance at most 2 from each other. It is known that if G has no cut-vertex, then G² is Hamilton-connected (see [G. Chartrand, A.M. Hobbs, H.A. Jung, S.F. Kapoor, C.St.J.A. Nash-Williams, The square of a block is hamiltonian connected, J. Combin. Theory Ser. B 16 (1974) 290-292; R.J. Faudree and R.H. Schelp, The square of a block is strongly path connected, J. Combin. Theory Ser. B 20 (1976) 47-61]). We prove that if G has only one cut-vertex, then G² is Hamilton-connected. In the case that G has only two cut-vertices, we prove that if the block that contains the two cut-vertices is hamiltonian, then G² is Hamilton-connected. Further, we characterize all graphs with at most one cycle having Hamilton-connected square.     

Hamilton cycles in claw-heavy graphs

A graph G on n≥3 vertices is called claw-heavy if every induced claw (K_1,3) of G has a pair of nonadjacent vertices such that their degree sum is at least n. In this paper we show that a claw-heavy graph G has a Hamilton cycle if we impose certain additional conditions on G involving numbers of common neighbors of some specific pair of nonadjacent vertices, or forbidden induced subgraphs. Our results extend two previous theorems of Broersma, Ryjá?ek and Schiermeyer [H.J. Broersma, Z. Ryjá?ek, I. Schiermeyer, Dirac’s minimum degree condition restricted to claws, Discrete Math. 167-168 (1997) 155-166], on the existence of Hamilton cycles in 2-heavy graphs.     

Transitive orientations in bull-reducible Berge graphs

A bull is a graph with five vertices r,y,x,z,s and five edges ry, yx, yz, xz, zs. A graph G is bull-reducible if every vertex of G lies in at most one bull of G. We prove that every bull-reducible Berge graph G that contains no antihole is weakly chordal, or has a homogeneous set, or is transitively orientable. This yields a fast polynomial time algorithm to color the vertices of such a graph exactly.     

Even and odd holes in cap‐free graphs

It is an old problem in graph theory to test whether a graph contains a chordless cycle of length greater than three (hole) with a specific parity (even, odd). Studying the structure of graphs without odd holes has obvious implications for Berge's strong perfect graph conjecture that states that a graph G is perfect if and only if neither G nor its complement contain an odd hole. Markossian, Gasparian, and Reed have proven that if neither G nor its complement contain an even hole, then G is β‐perfect. In this article, we extend the problem of testing whether G(V, E) contains a hole of a given parity to the case where each edge of G has a label odd or even. A subset of E is odd (resp. even) if it contains an odd (resp. even) number of odd edges. Graphs for which there exists a signing (i.e., a partition of E into odd and even edges) that makes every triangle odd and every hole even are called even‐signable. Graphs that can be signed so that every triangle is odd and every triangle is odd and every hole is odd are called odd‐signable. We derive from a theorem due to Truemper co‐NP characterizations of even‐signable and odd‐signable graphs. A graph is strongly even‐signable if it can be signed so that every cycle of length ≥ 4 with at most one chord is even and every triangle is odd. Clearly a strongly even‐signable graph is even‐signable as well. Graphs that can be signed so that cycles of length four with one chord are even and all other cycles with at most one chord are odd are called strongly odd‐signable. Every strongly odd‐signable graph is odd‐signable. We give co‐NP characterizations for both strongly even‐signable and strongly odd‐signable graphs. A cap is a hole together with a node, which is adjacent to exactly two adjacent nodes on the hole. We derive a decomposition theorem for graphs that contain no cap as induced subgraph (cap‐free graphs). Our theorem is analogous to the decomposition theorem of Burlet and Fonlupt for Meyniel graphs, a well‐studied subclass of cap‐free graphs. If a graph is strongly even‐signable or strongly odd‐signable, then it is cap‐free. In fact, strongly even‐signable graphs are those cap‐free graphs that are even‐signable. From our decomposition theorem, we derive decomposition results for strongly odd‐signable and strongly even‐signable graphs. These results lead to polynomial recognition algorithms for testing whether a graph belongs to one of these classes. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 289–308, 1999     

On powers of graphs of bounded NLC-width (clique-width)

Given a graph G, the graph G^l has the same vertex set and two vertices are adjacent in G^l if and only if they are at distance at most l in G. The l-coloring problem consists in finding an optimal vertex coloring of the graph G^l, where G is the input graph. We show that, for any fixed value of l, the l-coloring problem is polynomial when restricted to graphs of bounded NLC-width (or clique-width), if an expression of the graph is also part of the input. We also prove that the NLC-width of G^l is at most 2(l+1)^nlcw(G).     

On integral sum graphs

A graph G is said to be an integral sum graph if its nodes can be given a labeling f with distinct integers, so that for any two distinct nodes u and v of G, uv is an edge of G if and only if f(u)+f(v)=f(w) for some node w in G. A node of G is called a saturated node if it is adjacent to every other node of G. We show that any integral sum graph which is not K₃ has at most two saturated nodes. We determine the structure for all integral sum graphs with exactly two saturated nodes, and give an upper bound for the number of edges of a connected integral sum graph with no saturated nodes. We introduce a method of identification on constructing new connected integral sum graphs from given integral sum graphs with a saturated node. Moreover, we show that every graph is an induced subgraph of a connected integral sum graph. Miscellaneous related results are also presented.     

Edge-bipancyclicity of a hypercube with faulty vertices and edges

A bipartite graph G=(V,E) is said to be bipancyclic if it contains a cycle of every even length from 4 to |V|. Furthermore, a bipancyclic G is said to be edge-bipancyclic if every edge of G lies on a cycle of every even length. Let F_v (respectively, F_e) be the set of faulty vertices (respectively, faulty edges) in an n-dimensional hypercube Q_n. In this paper, we show that every edge of Q_n-F_v-F_e lies on a cycle of every even length from 4 to 2ⁿ-2|F_v| even if |F_v|+|F_e|?n-2, where n?3. Since Q_n is bipartite of equal-size partite sets and is regular of vertex-degree n, both the number of faults tolerated and the length of a longest fault-free cycle obtained are worst-case optimal.     

On embedding well-separable graphs

Call a simple graph H of order nwell-separable, if by deleting a separator set of size o(n) the leftover will have components of size at most o(n). We prove, that bounded degree well-separable spanning subgraphs are easy to embed: for every γ>0 and positive integer Δ there exists an n₀ such that if n>n₀, Δ(H)?Δ for a well-separable graph H of order n and δ(G)?(1-1/2(χ(H)-1)+γ)n for a simple graph G of order n, then H⊂G. We extend our result to graphs with small band-width, too.     

Finding large cycles in Hamiltonian graphs

We show how to find in Hamiltonian graphs a cycle of length n^{Ω(1/loglogn)}=exp(Ω(logn/loglogn)). This is a consequence of a more general result in which we show that if G has a maximum degree d and has a cycle with k vertices (or a 3-cyclable minor H with k vertices), then we can find in O(n³) time a cycle in G of length k^Ω(1/logd). From this we infer that if G has a cycle of length k, then one can find in O(n³) time a cycle of length k^{Ω(1/(log(n/k)+loglogn))}, which implies the result for Hamiltonian graphs. Our results improve, for some values of k and d, a recent result of Gabow (2004) [11] showing that if G has a cycle of length k, then one can find in polynomial time a cycle in G of length . We finally show that if G has fixed Euler genus g and has a cycle with k vertices (or a 3-cyclable minor H with k vertices), then we can find in polynomial time a cycle in G of length f(g)k^Ω(1), running in time O(n²) for planar graphs.     

Dominating sets and domatic number of circular arc graphs

A dominating set of a graph G = (N,E) is a subset S of nodes such that every node is either in S or adjacent to a node which is in S. The domatic number of G is the size of a maximum cardinality partition of N into dominating sets. The problems of finding a minimum cardinality dominating set and the domatic number are both NP-complete even for special classes of graphs. In the present paper we give an O(n∣E∣) time algorithm that finds a minimum cardinality dominating set when G is a circular arc graph (intersection graph of arcs on a circle). The domatic number problem is solved in O(n² log n) time when G is a proper circular arc graph, and it is shown NP-complete for general circular arc graphs.     

On 3-restricted edge connectivity of undirected binary Kautz graphs

An edge cut of a connected graph is m-restricted if its removal leaves every component having order at least m. The size of minimum m-restricted edge cuts of a graph G is called its m-restricted edge connectivity. It is known that when m≤4, networks with maximal m-restricted edge connectivity are most locally reliable. The undirected binary Kautz graph UK(2,n) is proved to be maximal 2- and 3-restricted edge connected when n≥3 in this work. Furthermore, every minimum 2-restricted edge cut disconnects this graph into two components, one of which being an isolated edge.     

The rank-width of the square grid

Rank-width is a graph width parameter introduced by Oum and Seymour. It is known that a class of graphs has bounded rank-width if, and only if, it has bounded clique-width, and that the rank-width of G is less than or equal to its branch-width.The n×nsquare grid, denoted by G_n,n, is a graph on the vertex set {1,2,…,n}×{1,2,…,n}, where a vertex (x,y) is connected by an edge to a vertex (x^′,y^′) if and only if |x−x^′|+|y−y^′|=1.We prove that the rank-width of G_n,n is equal to n−1, thus solving an open problem of Oum.     

Finding minimum dominating cycles in permutation graphs

A dominating cycle for a graph G = (V, E) is a subset C of V which has the following properties: (i) the subgraph of G induced by C has a Hamiltonian cycle, and (ii) every vertex of V is adjacent to some vertex of C. In this paper, we develop an O(n²) algorithm for finding a minimum cardinality dominating cycle in a permutation graph. We also show that a minimum cardinality dominating cycle in a permutation graph always has an even number of vertices unless it is isomorphic to C₃.     

Overlap number of graphs

An overlap representation of a graph G assigns sets to vertices so that vertices are adjacent if and only if their assigned sets intersect with neither containing the other. The overlap number φ(G) (introduced by Rosgen) is the minimum size of the union of the sets in such a representation. We prove the following: (1) An optimal overlap representation of a tree can be produced in linear time, and its size is the number of vertices in the largest subtree in which the neighbor of any leaf has degree 2. (2) If δ(G)?2 and G≠K₃, then φ(G)?|E(G)| ? 1, with equality when G is connected and triangle‐free and has no star‐cutset. (3) If G is an n‐vertex plane graph with n?5, then φ(G)?2n ? 5, with equality when every face has length 4 and there is no star‐cutset. (4) If G is an n‐vertex graph with n?14, then φ(G)?n²/4 ? n/2 ? 1, with equality for even n when G arises from K_{n/2, n/2} by deleting a perfect matching. © 2011 Wiley Periodicals, Inc. J Graph Theory     

Graphs with 1-Hamiltonian-connected cubes

The cube G³ of a connected graph G is that graph having the same vertex set as G and in which two distinct vertices are adjacent if and only if their distance in G is at most three. A Hamiltonian-connected graph has the property that every two distinct vertices are joined by a Hamiltonian path. A graph G is 1-Hamiltonian-connected if, for every vertex w of G, the graphs G and G?w are Hamiltonian-connected. A characterization of graphs whose cubes are 1-Hamiltonian-connected is presented.     

Planar unit-distance graphs having planar unit-distance complement

A graph G=(V,E) is called a unit-distance graph in the plane if there is an embedding of V into the plane such that every pair of adjacent vertices are at unit distance apart. If an embedding of V satisfies the condition that two vertices are adjacent if and only if they are at unit distance apart, then G is called a strict unit-distance graph in the plane. A graph G is a (strict) co-unit-distance graph, if both G and its complement are (strict) unit-distance graphs in the plane. We show by an exhaustive enumeration that there are exactly 69 co-unit-distance graphs (65 are strict co-unit-distance graphs), 55 of which are connected (51 are connected strict co-unit-distance graphs), and seven are self-complementary.