A graph G is said to be claw-free if G has no induced subgraph isomorphic to K1,3. For a cycle C in a graph G, C is called a Tutte cycle of G if C is a Hamilton cycle of G, or the order of C is at least 4 and every component of G − C has at most three neighbors on C. In [On a closure concept in claw-free graphs, J. Combin. Theory Ser. B 70 (1997), 217–224], Ryjáček proved that the conjectures b...

When searching for the longest cycle in a graph, we sometimes want to restrict our search space to the cycles that do not have crossing edges induced cycles. If we think of the crossing edges as shortcuts, looking for induced cycles is then looking for shortcut-free cycles. However, an induced cycle may have a shortcut in the following sense: For two vertices on the cycle, the distance between ...

The classic K-Cycle problem asks if a graph G, with vertex set V (G), has a simple cycle containing all vertices of a given set K TM V (G). In terms of colored graphs, it can be rephrased as follows: Given a graph G, a set K TM V (G) and an injective coloring c : K æ {1, 2, . . . , |K|}, decide if G has a simple cycle containing each color in {1, 2, . . . , |K|} (once). Another problem widely k...

A cycle decomposition of a graph Γ is a set C of cycles of Γ such that every edge of Γ belongs to exactly one cycle in C. Such a decomposition is called arc-transitive if the group of automorphisms of Γ that preserve C setwise acts transitively on the arcs of Γ . In this paper, we study arc-transitive cycle decompositions of tetravalent graphs. In particular, we are interested in determining an...

Graph bandwidth problem has been studies for over forty years. The relation between the bandwidth and number of edges in the graph is always interesting. The edge addition number ad(G) of a graph is the minimum number of edges that added into the graph G which cause the resulting graph’s bandwidth greater than the bandwidth of G. This paper determines the edge addition number for the Cartesian ...

Cai an Corneil (Discrete Math. 102 (1992) 103–106), proved that if a graph has a cycle double cover, then its line graph also has a cycle double cover, and that the validity of the cycle double cover conjecture on line graphs would imply the truth of the conjecture in general. In this note we investigate the conditions under which a graph G has a nowhere zero kow would imply that L(G), the line...

Let C2k denote the cycle of length 2k and Cg = {C3, C4, . . . , Cg}. The girth of a graph G containing a cycle is the length of a shortest cycle in G, so a graph has girth at least g + 1 if it is Cg-free. The distance between vertices u and v in a graph G, denoted dG(u, v) is the minimum length of a uv-path in G. Then dG(·, ·) is a metric on V (G). The diameter of a connected graph G is max{dG(...

An i-chord of a cycle or path is an edge whose endpoints are a distance i ≥ 2 apart along the cycle or path. Motivated by many standard graph classes being describable by the existence of chords, we investigate what happens when i-chords are required for specific values of i. Results include the following: A graph is strongly chordal if and only if, for i ∈ {4, 6}, every cycle C with |V (C)| ≥ ...

A graph G is bridged if every cycle C of length at least 4 has vertices x; y such that dG(x; y )< dC (x ; y ). A cycle C is isometric if dG(x; y )1⁄4 dC (x; y ) for all x; y 2 V (C). We show that every graph contractible to a graph with girth g has an isometric cycle of length at least g. We use this to show that every minimal cutset S in a bridged graph G induces a connected subgraph. We intro...

An even-cycle decomposition of a graph G is a partition of E(G) into cycles of even length. Evidently, every Eulerian bipartite graph has an even-cycle decomposition. Seymour (1981) proved that every 2-connected loopless Eulerian planar graph with an even number of edges also admits an even-cycle decomposition. Later, Zhang (1994) generalized this to graphs with no K5-minor. Our main theorem gi...