A geodominating set S ⊆ V of a graph G is said to be an odd geo-dominating if for every vertex , and The minimum cardinality the called geo-domination number denoted by . with – G. We develop algorithm compute graphs some families graphs.

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...