The Number of Out-Pancyclic Vertices in a Strong Tournament

An arc in a tournament T with n ≥ 3 vertices is called pancyclic, if it belongs to a cycle of length l for all 3 ≤ l ≤ n. We call a vertex u of T an out-pancyclic vertex of T, if each out-arc of u is pancyclic in T. Yao et al. (Discrete Appl. Math. 99, 245–249, 2000) proved that every strong tournament contains an out-pancyclic vertex. For strong tournaments with minimum out-degree 1, Yao et al. found an infinite class of strong tournaments, each of which contains exactly one out-pancyclic vertex. In this paper, we prove that every strong tournament with minimum out-degree at least 2 contains three out-pancyclic vertices. Our result is best possible since there is an infinite family of strong tournaments with minimum degree at least 2 and no more than 3 out-pancyclic vertices.     

Rooted Spanning Trees in Tournaments

A tournament of order n is an orientation of a complete graph with n vertices, and is specified by its vertex set V(T) and edge set E(T). A rooted tree is a directed tree such that every vertex except the root has in-degree 1, while the root has in-degree 0. A rooted k-tree is a rooted tree such that every vertex except the root has out-degree at most k; the out-degree of the root can be larger than k. It is well-known that every tournament contains a rooted spanning tree of depth at most 2; and the root of such a tree is also called a king in the literature. This result was strengthened to the following one: Every tournament contains a rooted spanning 2-tree of depth at most 2. We prove that every tournament of order n≥800 contains a spanning rooted special 2-tree in this paper, where a rooted special 2-tree is a rooted 2-tree of depth 2 such that all except possibly one, non-root, non-leaf vertices, have out-degree 2 in the tree. Revised: November 9, 1998     

Signless Laplacian spectral characterization of line graphs of T-shape trees

A T-shape tree is a tree with exactly one vertex of maximum degree 3. The line graphs of the T-shape trees are triangles with a hanging path at each vertex. Let C_a,b,c be such a graph, where a, b and c are the lengths of the paths. In this paper, we show that line graphs of T-shape trees, with the sole exception of C_a,a,2a+1, are determined by the spectra of their signless Laplacian matrices. For the graph C_a,a,2a+1 we identify the unique non-isomorphic graph sharing the same signless Laplacian characteristic polynomial.     

On tree‐decompositions of one‐ended graphs

A graph is one‐ended if it contains a ray (a one way infinite path) and whenever we remove a finite number of vertices from the graph then what remains has only one component which contains rays. A vertex v dominates a ray in the end if there are infinitely many paths connecting v to the ray such that any two of these paths have only the vertex v in common. We prove that if a one‐ended graph contains no ray which is dominated by a vertex and no infinite family of pairwise disjoint rays, then it has a tree‐decomposition such that the decomposition tree is one‐ended and the tree‐decomposition is invariant under the group of automorphisms. This can be applied to prove a conjecture of Halin from 2000 that the automorphism group of such a graph cannot be countably infinite and solves a recent problem of Boutin and Imrich. Furthermore, it implies that every transitive one‐ended graph contains an infinite family of pairwise disjoint rays.     

Graphs with homeomorphically irreducible spanning trees

It is an NP-complete problem to decide whether a graph contains a spanning tree with no vertex of degree 2. We show that these homeomorphically irreducible spanning trees are contained in all graphs with minimum degree at least c√n and in triangulations of the plane. They are nearly present in all graphs of diameter 2. They do not necessarily occur in r-regular or r-connected graphs.     

ON 3-CHOOSABILITY OF PLANE GRAPHS WITHOUT 6-,7- AND 9-CYCLES

The choice number of a graph G,denoted by X1(G),is the minimum number k such that if a list of k colors is given to each vertex of G,there is a vertex coloring of G where each vertex receives a color from its own list no matter what the lists are. In this paper,it is showed that X1 (G)≤3 for each plane graph of girth not less than 4 which contains no 6-, 7- and 9-cycles.     

Extremal cover times for random walks on trees

Let C_ν(T) denote the “cover time” of the tree T from the vertex v, that is, the expected number of steps before a random walk starting at v hits every vertex of T. Asymptotic lower bounds for C_ν(T) (for T a tree on n vertices) have been obtained recently by Kahn, Linial, Nisan and Saks, and by Devroye and Sbihi; here, we obtain the exact lower bound (approximately 2n In n) by showing that C_ν(T) is minimized when T is a star and v is one of its leaves. In addition, we show that the time to cover all vertices and then return to the starting point is minimized by a star (beginning at the center) and maximized by a path (beginning at one of the ends).     

The Four-in-a-Tree Problem in Triangle-Free Graphs

The three-in-a-tree algorithm of Chudnovsky and Seymour decides in time O(n ⁴) whether three given vertices of a graph belong to an induced tree. Here, we study four-in- a-tree for triangle-free graphs. We give a structural answer to the following question: what does a triangle-free graph look like if no induced tree covers four given vertices? Our main result says that any such graph must have the “same structure”, in a sense to be defined precisely, as a square or a cube. We provide an O(nm)-time algorithm that given a triangle-free graph G together with four vertices outputs either an induced tree that contains them or a partition of V(G) certifying that no such tree exists. We prove that the problem of deciding whether there exists a tree T covering the four vertices such that at most one vertex of T has degree at least 3 is NP-complete.     

Spanning trees in randomly perturbed graphs

A classical result of Komlós, Sárközy, and Szemerédi states that every n‐vertex graph with minimum degree at least (1/2 + o(1))n contains every n‐vertex tree with maximum degree . Krivelevich, Kwan, and Sudakov proved that for every n‐vertex graph G_α with minimum degree at least αn for any fixed α > 0 and every n‐vertex tree T with bounded maximum degree, one can still find a copy of T in G_α with high probability after adding O(n) randomly chosen edges to G_α. We extend the latter results to trees with (essentially) unbounded maximum degree; for a given and α > 0, we determine up to a constant factor the number of random edges that we need to add to an arbitrary n‐vertex graph with minimum degree αn in order to guarantee with high probability a copy of any fixed n‐vertex tree with maximum degree at most Δ.     

Subdivisions of Transitive Tournaments

We prove that, for r ≥ 2 andn ≥ n(r), every directed graph with n vertices and more edges than the r -partite Turán graph T(r, n) contains a subdivision of the transitive tournament on r + 1 vertices. Furthermore, the extremal graphs are the orientations ofT (r, n) induced by orderings of the vertex classes.     

Topological Graphs with No Large Grids

Let G be a topological graph with n vertices, i.e., a graph drawn in the plane with edges drawn as simple Jordan curves. It is shown that, for any constants k,l, there exists another constant C(k,l), such that if G has at least C(k,l)n edges, then it contains a k×l-gridlike configuration, that is, it contains k+l edges such that each of the first k edges crosses each of the last l edges. Moreover, one can require the first k edges to be incident to the same vertex. Received: April, 2003 Janos Pach and Micha Sharir has been supported by NSF Grants CCR-97-32101 and CCR-00-98246, and by a joint grant from the U.S.–Israel Binational Science Foundation. János Pach has also been supported by PSC-CUNY Research Award 63382-0032 and by OTKA T-032452. Micha Sharir has also been supported by a grant from the Israeli Academy of Sciences for a Center of Excellence in Geometric Computing at Tel Aviv University, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. Géza Tóth has been supported by OTKA-T-038397 and by an award from the New York University Research Challenge Fund.     

Claw‐free graphs and 2‐factors that separate independent vertices

In this article, we prove that a line graph with minimum degree δ≥7 has a spanning subgraph in which every component is a clique of order at least three. This implies that if G is a line graph with δ≥7, then for any independent set S there is a 2‐factor of G such that each cycle contains at most one vertex of S. This supports the conjecture that δ≥5 is sufficient to imply the existence of such a 2‐factor in the larger class of claw‐free graphs. It is also shown that if G is a claw‐free graph of order n and independence number α with δ≥2n/α?2 and n≥3α³/2, then for any maximum independent set S, G has a 2‐factor with α cycles such that each cycle contains one vertex of S. This is in support of a conjecture that δ≥n/α≥5 is sufficient to imply the existence of a 2‐factor with α cycles, each containing one vertex of a maximum independent set. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 251–263, 2012     

On the existence of special depth first search trees

The Depth First Search (DFS) algorithm is one of the basic techniques that is used in a very large variety of graph algorithms. Most applications of the DFS involve the construction of a depth-first spanning tree (DFS tree). In this paper, we give a complete characterization of all the graphs in which every spanning tree is a DFS tree. These graphs are called Total-DFS-Graphs. We prove that Total-DFS-Graphs are closed under minors. It follows by the work of Robertson and Seymour on graph minors that there is a finite set of forbidden minors of these graphs and that there is a polynomial algorithm for their recognition. We also provide explicit characterizations of these graphs in terms of forbidden minors and forbidden topological minors. The complete characterization implies explicit linear algorithm for their recognition. In some problems the degree of some vertices in the DFS tree obtained in a certain run are crucial and therefore we also consider the following problem: Let G = (V,E) be a connected undirected graph where |V| = n and let d ? Nⁿ be a degree sequence upper bound vector. Is there any DFS tree T with degree sequence d _T that violates d (i.e., d _T ≤ d , which means: E j such that d _T(j) > d (j))? We show that this problem is NP-complete even for the case where we restrict the degree of only on specific vertex to be less than or equal to k for a fixed k ≥ 2 (i.e., d = (n - 1, ?, n - 1, k, n - 1, ?, n - 1)). 0 1995 John Wiley & Sons, Inc.     

An edge-deletion problem for locally finite graphs

For each positive integer n, let T_n be the tree in which exactly one vertex has degree n and all the other vertices have degree n + 1. A graph G is called stable if its edge set is nonempty and if deleting an arbitrary edge of G there is always a component of the residue graph which is isomorphic to G. The question whether there are locally finite stable graphs that are not isomorphic to one of the graphs T_n is answered affirmatively by constructing an uncountable family of pairwise nonisomorphic, locally finite, stable graphs. Further, the following results are proved: (1) Among the locally finite trees containing no subdivision of T₂, the oneway infinite path T₁ is the only stable graph. (2) Among the locally finite graphs containing no two-way infinite path, T₁ is also the only stable graph.     

Degree conditions for 2-factors

For any positive integer k, we investigate degree conditions implying that a graph G of order n contains a 2-factor with exactly k components (vertex disjoint cycles). In particular, we prove that for k ≤ (n/4), Ore's classical condition for a graph to be hamiltonian (k = 1) implies that the graph contains a 2-factor with exactly k components. We also obtain a sufficient degree condition for a graph to have k vertex disjoint cycles, at least s of which are 3-cycles and the remaining are 4-cycles for any s ≤ k. © 1997 John Wiley & Sons, Inc.     

Convex labelings of trees

A convex labeling of a tree T of order n is a one-to-one function f from the vertex set of T into the nonnegative integers, so that f(y) ? (f(x) + f(z))/2 for every path x, y, z of length 2 in T. If, in addition, f(v) ? n ? 1 for every vertex v of T, then f is a perfect convex labeling and T is called a perfectly convex tree. Jamison introduced this concept and conjectured that every tree is perfectly convex. We show that there exists an infinite class of trees, none of which is perfectly convex, and in fact prove that for every n there exists a tree of order n which requires a convex labeling with maximum value at least 6n/5 – 22. We also prove that every tree of order n admits a convex labeling with maximum label no more than n²/8 + 2. In addition, we present some constructive methods for obtaining perfect convex labelings of large classes of trees.     

The 2-Pebbling Property and a Conjecture of Graham's

The pebbling number of a graph G, f(G), is the least m such that, however m pebbles are placed on the vertices of G, we can move a pebble to any vertex by a sequence of moves, each move taking two pebbles off one vertex and placing one on an adjacent vertex. It is conjectured that for all graphs G and H, f(G 2H)hf(G)f(H).¶Let C_m and C_n be cycles. We prove that f(C_m 2C_n)hf(C_m) f(C_n) for all but a finite number of possible cases. We also prove that f(G2T)hf(G) f(T) when G has the 2-pebbling property and T is any tree.     

On the largest tree of given maximum degree in a connected graph

We prove that every connected graph G contains a tree T of maximum degree at most k that either spans G or has order at least kδ(G) + 1, where δ(G) is the minimum degree of G. This generalizes and unifies earlier results of Bermond [1] and Win [7]. We also show that the square of a connected graph contains a spanning tree of maximum degree at most three.     

Four‐cycles in graphs without a given even cycle

We prove that every bipartite C_2l‐free graph G contains a C₄‐free subgraph H with e(H) ≥ e(G)/(l – 1). The factor 1/(l – 1) is best possible. This implies that ex(n, C_2l) ≤ 2(l – 1)ex(n, {C₄, C_2l}), which settles a special case of a conjecture of Erd?s and Simonovits. © 2004 Wiley Periodicals, Inc. J Graph Theory 48: 147–156, 2005     

On the number of complementary trees in a graph

Consider a graph with no loops or multiple arcs with n+1 nodes and 2n arcs labeled a_l,…,a_n,a′_l,…,a′_n, where n ≥ 5. A spanning tree of such a graph is called complementary if it contains exactly one arc of each pair {a_i,a′_i}. The purpose of this paper is to develop a procedure for finding complementary trees in a graph, given one such tree. Using the procedure repeatedly we give a constructive proof that every graph of the above form which has one complementary tree has at least six such trees.