We observe that for any logarithmically concave nite sequence a 0 , a 1 , : : : , a n of positive integers there is a representation of the Lie algebra sl 2 (C) from which this logarithmic concav-ity follows. Thus, in applying this strategy to prove logarithmic concavity, the only issue is to construct such a representation naturally from given combinatorial data. As an example, we do this when...

In this paper, we consider the circular chromatic number c (G) of series-parallel graphs G. It is well known that series-parallel graphs have chromatic number at most 3. Hence their circular chromatic number is also at most 3. If a series-parallel graph G contains a triangle , then both the chromatic number and the circular chromatic number of G are indeed equal to 3. We shall show that if a se...

A graph G is called triangle-free if G has no induced K3 as a subgraph. We set 3 = min{3i=1 d(vi)|{v1; v2; v3} is an independent set of vertices in G}. In this paper, we show that if G is a 1-tough and triangle-free graph of order n with n6 3, then G is hamiltonian. c © 2002 Elsevier Science B.V. All rights reserved.

The circumference of a graph is the length of its longest cycles. Results of Jackson, and Jackson and Wormald, imply that the circumference of a 3-connected cubic n-vertex graph is Ω(n), and the circumference of a 3-connected claw-free graph is Ω(n). We generalise and improve the first result by showing that every 3-edge-connected graph with m edges has an Eulerian subgraph with Ω(m) edges. We ...

Let γ(G) be the domination number of a graph G. A graph G is domination-vertex-critical, or γ-vertex-critical, if γ(G− v) < γ(G) for every vertex v ∈ V (G). In this paper, we show that: Let G be a γ-vertex-critical graph and γ(G) = 3. (1) If G is of even order and K1,6-free, then G has a perfect matching; (2) If G is of odd order and K1,7-free, then G has a near perfect matching with only three...

Thomassen conjectured that every 4-connected line graph is Hamiltonian. A vertex cut X of G is essential if G−X has at least two non-trivial components. We prove that every 3-connected, essentially 11-connected line graph is Hamiltonian. Using Ryjác̆ek’s line graph closure, it follows that every 3-connected, essentially 11-connected claw-free graph is Hamiltonian. © 2005 Elsevier Inc. All rights...

We show that for a given set S of 2n points in general position in the plane, there is a cubic 3-connected crossing-free geometric graph on the vertex set S if and only if the interior of the convex hull of S contains at least n − 1 points of S. We also give similar results for sets of odd cardinality. Analogous partial results are given if the 3-connected crossing-free geometric graph is not r...

We consider the question of the range of the number of cycles possible in a 2-factor of a 2-connected claw-free graph with sufficiently high minimum degree. (By claw-free we mean the graph has no induced K1,3.) In particular, we show that for such a graph G of order n ≥ 51 with δ(G) ≥ n−2 3 , G contains a 2-factor with exactly k cycles, for 1 ≤ k ≤ n−24 3 . We also show that this result is shar...

Let k ≥ 4, and let H1, H2 be connected graphs with |V (Hi)| ≥ 3 for i = 1, 2. A graph G is said to be {H1, . . . , Hl}-free if none of H1, . . . , Hl is an induced subgraph of G. We prove that if there exists a positive integer n0 such that every {H1, H2}-free graph G with |V (G)| ≥ n0 and δ(G) ≥ 3 contains k vertex-disjoint claws, then {H1, H2} ∩ {K1,t | t ≥ 2} = ∅. Also, we prove that every K...

Matthews and Sumner have proved in [10] that if G is a 2-connected claw-free graph of order n such that δ(G) ≥ (n − 2)/3, then G is Hamiltonian. We say that a graph is almost claw-free if for every vertex v of G, 〈N(v)〉 is 2-dominated and the set A of centers of claws of G is an independent set. Broersma et al. [5] have proved that if G is a 2-connected almost claw-free graph of order n such th...