Let G = (V, E) be a graph and a weight function w : E -+ Z+. Let T C V be an even subset of the vertices of G. A T-cut is an edge-cutset of the graph which divides T into two odd sets. A T-join is a minimal subset of edges that meets every T-cut (a generalization of solutionS to the Chinese postman problem). The main theorem of this paper gives a tight upper bound on the difference between the ...

Gyárfás conjectured in 1985 that for all k, l, every graph with no clique of size more than k and no odd hole of length more than l has chromatic number bounded by a function of k, l. We prove three weaker statements: • Every triangle-free graph with sufficiently large chromatic number has an odd hole of length different from five; • For all l, every triangle-free graph with sufficiently large ...

Finding edge-disjoint odd cycles is one of the most important problems in graph theory, graph algorithm and combinatorial optimization. In fact, it is closely related to the well-known max-cut problem. One of the difficulties of this problem is that the Erdős-Pósa property does not hold for odd cycles in general. Motivated by this fact, we prove that for any positive integer k, there exists an ...

An extremal graph for a graph H on n vertices is a graph on n vertices with maximum number of edges that does not contain H as a subgraph. Let Tn,r be the Turán graph, which is the complete r-partite graph on n vertices with part sizes that differ by at most one. The well-known Turán Theorem states that Tn,r is the only extremal graph for complete graph Kr+1. Erdős et al. (1995) determined the ...

In a drawing of a graph, two edges form an odd pair if they cross each other an odd number of times. A pair of edges is independent if they share no endpoint. For a graph G, let ocr(G) be the smallest number of odd pairs in a drawing of G and let iocr(G) be the smallest number of independent odd pairs in a drawing of G. We construct a graph G with iocr(G) < ocr(G), answering a question by Széke...

Gyárfás conjectured in 1985 that for all k, l, every graph with no clique of size more than k and no odd hole of length more than l has chromatic number bounded by a function of k, l. We prove three weaker statements: • Every triangle-free graph with sufficiently large chromatic number has an odd hole of length different from five; • For all l, every triangle-free graph with sufficiently large ...

A graph is arc-regular if its automorphism group acts sharply-transitively on the set of its ordered edges. This paper answers an open question about the existence of arc-regular 3-valent graphs of order 4m where m is an odd integer. Using the Gorenstein–Walter theorem, it is shown that any such graph must be a normal cover of a base graph, where the base graph has an arc-regular group of autom...

Given a graph G, the acyclic orientation graph of G, denoted AO(G), is the graph whose vertices are the acyclic orientations of G, and two acyclic orientations are joined by an edge in AO(G) iff they differ by the reversal of a single edge. A hamilton cycle in AO(G) gives a Gray code listing of the acyclic orientations of G. We prove that for certain graphs G, AO(G) is hamiltonian, and give exp...