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5 results on '"ore-type condition"'

1. Heavy subgraph pairs for traceability of block-chains

Author

Li Binlong, Broersma Hajo, and Zhang Shenggui

Subjects
o−1-heavy subgraph, block-chain traceable graph, ore-type condition, forbidden subgrap, Mathematics, QA1-939
Abstract

A graph is called traceable if it contains a Hamilton path, i.e., a path containing all its vertices. Let G be a graph on n vertices. We say that an induced subgraph of G is o−1-heavy if it contains two nonadjacent vertices which satisfy an Ore-type degree condition for traceability, i.e., with degree sum at least n−1 in G. A block-chain is a graph whose block graph is a path, i.e., it is either a P1, P2, or a 2-connected graph, or a graph with at least one cut vertex and exactly two end-blocks. Obviously, every traceable graph is a block-chain, but the reverse does not hold. In this paper we characterize all the pairs of connected o−1-heavy graphs that guarantee traceability of block-chains. Our main result is a common extension of earlier work on degree sum conditions, forbidden subgraph conditions and heavy subgraph conditions for traceability

Published
2014
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2. Heavy subgraph pairs for traceability of block-chains

Author

Li, Binlong, Broersma, Haitze J., Zhang, Shenggui, and Discrete Mathematics and Mathematical Programming

Subjects
Factor-critical graph, EWI-24776, o−1-heavy subgraph, Induced subgraph, Combinatorics, Ore-type condition, MSC-05C45, QA1-939, Discrete Mathematics and Combinatorics, Traceable graph, Forbidden graph characterization, Universal graph, Mathematics, Distance-hereditary graph, MSC-05C07, Discrete mathematics, Applied Mathematics, Neighbourhood (graph theory), Block-chain, Forbidden subgraph, forbidden subgrap, $o_{-1}$-Heavy subgraph, METIS-304107, block-chain traceable graph, Path graph, MSC-05C38, Graph factorization, IR-91190
Abstract

A graph is called traceable if it contains a Hamilton path, i.e., a path containing all its vertices. Let G be a graph on n vertices. We say that an induced subgraph of G is o-1-heavy if it contains two nonadjacent vertices which satisfy an Ore-type degree condition for traceability, i.e., with degree sum at least n−1 in G. A block-chain is a graph whose block graph is a path, i.e., it is either a P1, P2, or a 2-connected graph, or a graph with at least one cut vertex and exactly two end-blocks. Obviously, every traceable graph is a block-chain, but the reverse does not hold. In this paper we characterize all the pairs of connected o-1-heavy graphs that guarantee traceability of block-chains. Our main result is a common extension of earlier work on degree sum conditions, forbidden subgraph conditions and heavy subgraph conditions for traceability.

Published
2014

4. Matchings, factors and cycles in graphs

Author

Philpotts, Adam Richard and Philpotts, Adam Richard

Abstract

A matching in a graph is a set of pairwise nonadjacent edges, a k-factor is a k-regular spanning subgraph, and a cycle is a closed path. This thesis has two parts. In Part I (by far the larger part) we study sufficient conditions for structures involving matchings, factors and cycles. The three main types of conditions involve: the minimum degree; the degree sum of pairs of nonadjacent vertices (Ore-type conditions); and the neighbourhoods of independent sets of vertices. We show that most of our theorems are best possible by giving appropriate extremal graphs. We study Ore-type conditions for a graph to have a Hamilton cycle or 2-factor containing a given matching or path-system, and for any matching and single vertex to be contained in a cycle. We give Ore-type and neighbourhood conditions for a matching L of l edges to be contained in a matching of k edges (l < k). We generalise two different aspects of this result: the l = 0 case with an Ore-type condition for a heavy matching in an edge-weighted graph; and the conditions for a perfect matching containing L with degree and neighbourhood conditions for a k-factor (k > 2) containing a given set of edges. We also establish neighbourhood conditions for the existence of a cycle of length at least k. A list-edge-colouring of a graph is an assignment of a colour to each edge from its own list of colours. In Part II we study edge colourings of powers of cycles, and prove the List-Edge-Colouring Conjecture for squares of cycles of odd length.

5. Matchings, factors and cycles in graphs

Author

Philpotts, Adam Richard and Philpotts, Adam Richard

Abstract

A matching in a graph is a set of pairwise nonadjacent edges, a k-factor is a k-regular spanning subgraph, and a cycle is a closed path. This thesis has two parts. In Part I (by far the larger part) we study sufficient conditions for structures involving matchings, factors and cycles. The three main types of conditions involve: the minimum degree; the degree sum of pairs of nonadjacent vertices (Ore-type conditions); and the neighbourhoods of independent sets of vertices. We show that most of our theorems are best possible by giving appropriate extremal graphs. We study Ore-type conditions for a graph to have a Hamilton cycle or 2-factor containing a given matching or path-system, and for any matching and single vertex to be contained in a cycle. We give Ore-type and neighbourhood conditions for a matching L of l edges to be contained in a matching of k edges (l < k). We generalise two different aspects of this result: the l = 0 case with an Ore-type condition for a heavy matching in an edge-weighted graph; and the conditions for a perfect matching containing L with degree and neighbourhood conditions for a k-factor (k > 2) containing a given set of edges. We also establish neighbourhood conditions for the existence of a cycle of length at least k. A list-edge-colouring of a graph is an assignment of a colour to each edge from its own list of colours. In Part II we study edge colourings of powers of cycles, and prove the List-Edge-Colouring Conjecture for squares of cycles of odd length.

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