Chordal 2‐Connected Graphs and Spanning Trees

We present a transformation on a chordal 2‐connected simple graph that decreases the number of spanning trees. Based on this transformation, we show that for positive integers n, m with , the threshold graph having n vertices and m edges that consists of an ‐clique and vertices of degree 2 is the only graph with the fewest spanning trees among all 2‐connected chordal graphs on n vertices and m edges.     

A Density Turán Theorem

Let F be a graph that contains an edge whose deletion reduces its chromatic number. For such a graph F , a classical result of Simonovits from 1966 shows that every graph on vertices with more than edges contains a copy of F . In this article we derive a similar theorem for multipartite graphs. For a graph H and an integer , let be the minimum real number such that every ?‐partite graph whose edge density between any two parts is greater than contains a copy of H . Our main contribution in this article is to show that for all sufficiently large if and only if H admits a vertex‐coloring with colors such that all color classes but one are independent sets, and the exceptional class induces just a matching. When H is a complete graph, this recovers a result of Pfender (Combinatorica 32 (2012), 483–495). We also consider several extensions of Pfender's result.     

Hadwiger's Conjecture for the Complements of Kneser Graphs

Hadwiger's conjecture asserts that every graph with chromatic number t contains a complete minor of order t. Given integers , the Kneser graph is the graph with vertices the k‐subsets of an n‐set such that two vertices are adjacent if and only if the corresponding k‐subsets are disjoint. We prove that Hadwiger's conjecture is true for the complements of Kneser graphs.     

Expression for the Number of Spanning Trees of Line Graphs of Arbitrary Connected Graphs

For any graph G, let be the number of spanning trees of G, be the line graph of G, and for any nonnegative integer r, be the graph obtained from G by replacing each edge e by a path of length connecting the two ends of e. In this article, we obtain an expression for in terms of spanning trees of G by a combinatorial approach. This result generalizes some known results on the relation between and and gives an explicit expression if G is of order and size in which s vertices are of degree 1 and the others are of degree k. Thus we prove a conjecture on for such a graph G.     

Counterexamples to the List Square Coloring Conjecture

The square G² of a graph G is the graph defined on such that two vertices u and v are adjacent in G² if the distance between u and v in G is at most 2. Let and be the chromatic number and the list chromatic number of a graph H, respectively. A graph H is called chromatic‐choosable if . It is an interesting problem to find graphs that are chromatic‐choosable. Kostochka and Woodall (Choosability conjectures and multicircuits, Discrete Math., 240 (2001), 123–143) conjectured that for every graph G, which is called List Square Coloring Conjecture. In this article, we give infinitely many counter examples to the conjecture. Moreover, we show that the value can be arbitrarily large.     

Vizing's 2‐Factor Conjecture Involving Large Maximum Degree

Let G be an n‐vertex simple graph, and let and denote the maximum degree and chromatic index of G, respectively. Vizing proved that or . Define G to be Δ‐critical if and for every proper subgraph H of G. In 1965, Vizing conjectured that if G is an n‐vertex Δ‐critical graph, then G has a 2‐factor. Luo and Zhao showed if G is an n‐vertex Δ‐critical graph with , then G has a hamiltonian cycle, and so G has a 2‐factor. In this article, we show that if G is an n‐vertex Δ‐critical graph with , then G has a 2‐factor.     

On Meyniel's conjecture of the cop number

Meyniel conjectured that the cop number c(G) of any connected graph G on n vertices is at most for some constant C. In this article, we prove Meyniel's conjecture in special cases that G has diameter 2 or G is a bipartite graph of diameter 3. For general connected graphs, we prove , improving the best previously known upper‐bound O(n/ lnn) due to Chiniforooshan.     

On the Minimum Number of Spanning Trees in k‐Edge‐Connected Graphs

We show that a k‐edge‐connected graph on n vertices has at least spanning trees. This bound is tight if k is even and the extremal graph is the n‐cycle with edge multiplicities . For k odd, however, there is a lower bound , where . Specifically, and . Not surprisingly, c₃ is smaller than the corresponding number for 4‐edge‐connected graphs. Examples show that . However, we have no examples of 5‐edge‐connected graphs with fewer spanning trees than the n‐cycle with all edge multiplicities (except one) equal to 3, which is almost 6‐regular. We have no examples of 5‐regular 5‐edge‐connected graphs with fewer than spanning trees, which is more than the corresponding number for 6‐regular 6‐edge‐connected graphs. The analogous surprising phenomenon occurs for each higher odd edge connectivity and regularity.     

Hadwiger's Conjecture for ℓ‐Link Graphs

In this article, we define and study a new family of graphs that generalizes the notions of line graphs and path graphs. Let G be a graph with no loops but possibly with parallel edges. An ?‐link of G is a walk of G of length in which consecutive edges are different. The ?‐link graph of G is the graph with vertices the ?‐links of G , such that two vertices are joined by edges in if they correspond to two subsequences of each of μ ‐links of G . By revealing a recursive structure, we bound from above the chromatic number of ?‐link graphs. As a corollary, for a given graph G and large enough ?, is 3‐colorable. By investigating the shunting of ?‐links in G , we show that the Hadwiger number of a nonempty is greater or equal to that of G . Hadwiger's conjecture states that the Hadwiger number of a graph is at least the chromatic number of that graph. The conjecture has been proved by Reed and Seymour (Eur J Combin 25(6) (2004), 873–876) for line graphs, and hence 1‐link graphs. We prove the conjecture for a wide class of ?‐link graphs.     

Antimagic labelling of vertex weighted graphs

Suppose G is a graph, k is a non‐negative integer. We say G is k‐antimagic if there is an injection f: E→{1, 2, …, |E| + k} such that for any two distinct vertices u and v, . We say G is weighted‐k‐antimagic if for any vertex weight function w: V→?, there is an injection f: E→{1, 2, …, |E| + k} such that for any two distinct vertices u and v, . A well‐known conjecture asserts that every connected graph G≠K₂ is 0‐antimagic. On the other hand, there are connected graphs G≠K₂ which are not weighted‐1‐antimagic. It is unknown whether every connected graph G≠K₂ is weighted‐2‐antimagic. In this paper, we prove that if G has a universal vertex, then G is weighted‐2‐antimagic. If G has a prime number of vertices and has a Hamiltonian path, then G is weighted‐1‐antimagic. We also prove that every connected graph G≠K₂ on n vertices is weighted‐ ?3n/2?‐antimagic. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory     

On Hypohamiltonian and Almost Hypohamiltonian Graphs

A graph G is almost hypohamiltonian if G is non‐hamiltonian, there exists a vertex w such that is non‐hamiltonian, and for any vertex the graph is hamiltonian. We prove the existence of an almost hypohamiltonian graph with 17 vertices and of a planar such graph with 39 vertices. Moreover, we find a 4‐connected almost hypohamiltonian graph, while Thomassen's question whether 4‐connected hypohamiltonian graphs exist remains open. We construct planar almost hypohamiltonian graphs of order n for every . During our investigation we draw connections between hypotraceable, hypohamiltonian, and almost hypohamiltonian graphs, and discuss a natural extension of almost hypohamiltonicity. Finally, we give a short argument disproving a conjecture of Chvátal (originally disproved by Thomassen), strengthen a result of Araya and Wiener on cubic planar hypohamiltonian graphs, and mention open problems.     

More on foxes

An edge in a k‐connected graph G is called k‐contractible if the graph obtained from G by contracting e is k‐connected. Generalizing earlier results on 3‐contractible edges in spanning trees of 3‐connected graphs, we prove that (except for the graphs if ) (a) every spanning tree of a k‐connected triangle free graph has two k‐contractible edges, (b) every spanning tree of a k‐connected graph of minimum degree at least has two k‐contractible edges, (c) for , every DFS tree of a k‐connected graph of minimum degree at least has two k‐contractible edges, (d) every spanning tree of a cubic 3‐connected graph nonisomorphic to K₄ has at least many 3‐contractible edges, and (e) every DFS tree of a 3‐connected graph nonisomorphic to K₄, the prism, or the prism plus a single edge has two 3‐contractible edges. We also discuss in which sense these theorems are best possible.     

Pebbling Graphs of Diameter Three and Four

Given a configuration of pebbles on the vertices of a connected graph G, a pebbling move is defined as the removal of two pebbles from some vertex and the placement of one of these on an adjacent vertex. The pebbling number of a graph G is the smallest integer k such that for each vertex v and each configuration of k pebbles on G there is a sequence of pebbling moves that places at least one pebble on v. We improve on a bound of Bukh by showing that the pebbling number of a graph of diameter three on n vertices is at most , and this bound is best possible. Further, we obtain an asymptotic bound of for the pebbling number of graphs of diameter four. Finally, we prove an asymptotic bound for pebbling graphs of arbitrary diameter, namely that the pebbling number for a diameter d graph on n vertices is at most , where is a constant depending upon d. This also improves another bound of Bukh.     

Nonseparating K4‐subdivisions in graphs of minimum degree at least 4

We first prove that for every vertex x of a 4‐connected graph G, there exists a subgraph H in G isomorphic to a subdivision of the complete graph K₄ on four vertices such that is connected and contains x. This implies an affirmative answer to a question of Kühnel whether every 4‐connected graph G contains a subdivision H of K₄ as a subgraph such that is connected. The motor for our induction is a result of Fontet and Martinov stating that every 4‐connected graph can be reduced to a smaller one by contracting a single edge, unless the graph is the square of a cycle or the line graph of a cubic graph. It turns out that this is the only ingredient of the proof where 4‐connectedness is used. We then generalize our result to connected graphs of minimum degree at least 4 by developing the respective motor, a structure theorem for the class of simple connected graphs of minimum degree at least 4. A simple connected graph G of minimum degree 4 cannot be reduced to a smaller such graph by deleting a single edge or contracting a single edge and simplifying if and only if it is the square of a cycle or the edge disjoint union of copies of certain bricks as follows: Each brick is isomorphic to K₃, K₅, K_{2, 2, 2}, , , or one the four graphs , , , obtained from K₅ and K_{2, 2, 2} by deleting the edges of a triangle, or replacing a vertex x by two new vertices and adding four edges to the endpoints of two disjoint edges of its former neighborhood, respectively. Bricks isomorphic to K₅ or K_{2, 2, 2} share exactly one vertex with the other bricks of the decomposition, vertices of degree 4 in any other brick are not contained in any further brick of the decomposition, and the vertices of a brick isomorphic to K₃ must have degree 4 in G and have pairwise no common neighbors outside that brick.     

Gomory‐Hu trees of infinite graphs with finite total weight

A well‐known theorem of Gomory and Hu states that if G is a finite graph with nonnegative weights on its edges, then there exists a tree T (now called a Gomory‐Hu tree) on such that for all there is an such that the two components of determine an optimal (minimal valued) cut between u an v in G. In this article, we extend their result to infinite weighted graphs with finite total weight. Furthermore, we show by an example that one cannot omit the condition of the finiteness of the total weight.     

On a Conjecture of Erdős,Gallai, and Tuza

Erd?s, Gallai, and Tuza posed the following problem: given an n‐vertex graph G, let denote the smallest size of a set of edges whose deletion makes G triangle‐free, and let denote the largest size of a set of edges containing at most one edge from each triangle of G. Is it always the case that ? We have two main results. We first obtain the upper bound , as a partial result toward the Erd?s–Gallai–Tuza conjecture. We also show that always , where m is the number of edges in G; this bound is sharp in several notable cases.     

A generalization of Gale's lemma

In this work, we present a generalization of Gale's lemma. Using this generalization, we introduce two sharp combinatorial lower bounds for and , the two classic topological lower bounds for the chromatic number of a graph G.     

Diameter of random spanning trees in a given graph

Motivated by the observation that the sparse tree‐like subgraphs in a small world graph have large diameter, we analyze random spanning trees in a given host graph. We show that the diameter of a random spanning tree of a given host graph G is between and with high probability., where c and c′ depend on the spectral gap of G and the ratio of the moments of the degree sequence. For the special case of regular graphs, this result improves the previous lower bound by Aldous by a factor of logn. Copyright © 2011 John Wiley Periodicals, Inc. J Graph Theory 69: 223–240, 2012     

A greedy algorithm for finding a large 2‐matching on a random cubic graph

A 2‐matching of a graph G is a spanning subgraph with maximum degree two. The size of a 2‐matching U is the number of edges in U and this is at least where n is the number of vertices of G and κ denotes the number of components. In this article, we analyze the performance of a greedy algorithm 2greedy for finding a large 2‐matching on a random 3‐regular graph. We prove that with high probability, the algorithm outputs a 2‐matching U with .     

Degree sum and vertex dominating paths

A vertex dominating path in a graph is a path P such that every vertex outside P has a neighbor on P. In 1988 H. Broersma [5] stated a result implying that every n‐vertex k‐connected graph G such that contains a vertex dominating path. We provide a short, self‐contained proof of this result and further show that every n‐vertex k‐connected graph such that contains a vertex dominating path of length at most , where T is a minimum dominating set of vertices. An immediate corollary of this result is that every such graph contains a vertex dominating path with length bounded above by a logarithmic function of the order of the graph. To derive this result, we prove that every n‐vertex k‐connected graph with contains a path of length at most , through any set of T vertices where .