Proper‐walk connection number of graphs

Bang‐Jensen, Jørgen; Bellitto, Thomas; Yeo, Anders

doi:10.1002/jgt.22609

Cited by 7 publications

(4 citation statements)

References 20 publications

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“…In addition to the connectivity, we know that graphs have many other global structural properties, such as spanning trees, Hamiltonian cycles (paths) and so on. Similarly, combining them with graph colorings, one may think about introducing some other new global colorings and global chromatic numbers, for example, proper-walk connection coloring and number in a recent paper [6].…”

Section: Colored Disconnection Coloringsmentioning

confidence: 99%

Graph colorings under global structural conditions

Bai,

2020

Preprint

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More than ten years ago in 2008, a new kind of graph coloring appeared in graph theory, which is the rainbow connection coloring of graphs, and then followed by some other new concepts of graph colorings, such as proper connection coloring, monochromatic connection coloring, and conflict-free connection coloring of graphs. In about ten years of our consistent study, we found that these new concepts of graph colorings are actually quite different from the classic graph colorings. These colored connection colorings of graphs are brand-new colorings and they need to take care of global structural properties (for example, connectivity) of a graph under the colorings; while the traditional colorings of graphs are colorings under which only local structural properties (adjacent vertices or edges) of a graph are taken care of. Both classic colorings and the new colored connection colorings can produce the so-called chromatic numbers. We call the colored connection numbers the global chromatic numbers, and the classic or traditional chromatic numbers the local chromatic numbers. This paper intends to clarify the difference between the colored connection colorings and the traditional colorings, and finally to propose the new concepts of global colorings under which global structural properties of the colored graph are kept, and the global chromatic numbers.

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Section: Colored Disconnection Coloringsmentioning

confidence: 99%

Graph colorings under global structural conditions

Bai,

2020

Preprint

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“…This has been used several times in the literature, see e.g. [1,15] and [5,Chapter 16]. Let G = (X, Y, E) be a bipartite graph for which each edge is coloured red or blue.…”

Section: Alternating Cycles In 2-edge-coloured Graphsmentioning

confidence: 99%

Supereulerian 2-edge-coloured graphs

Bang‐Jensen

Bellitto

Yeo

2020

Preprint

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A 2-edge-coloured graph G is supereulerian if G contains a spanning closed trail in which the edges alternate in colours. An eulerian factor of a 2-edge-coloured graph is a collection of vertex disjoint induced subgraphs which cover all the vertices of G such that each of these subgraphs is supereulerian. We give a polynomial algorithm to test if a 2-edge-coloured graph has an eulerian factor and to produce one when it exists. A 2edge-coloured graph is (trail-)colour-connected if it contains a pair of alternating (u, v)-paths ((u, v)-trails) whose union is an alternating closed walk for every pair of distinct vertices u, v. A 2-edge-coloured graph is M-closed if xz is an edge of G whenever some vertex u is joined to both x and z by edges of the same colour. M-closed 2-edge-coloured graphs, introduced in [11], form a rich generalization of 2-edge-coloured complete graphs. We show that if G is an extension of an M-closed 2-edge-coloured complete graph, then G is supereulerian if and only if G is trail-colourconnected and has an eulerian factor. We also show that for general 2-edge-coloured graphs it is NP-complete to decide whether the graph is supereulerian. Finally we pose a number of open problems. Keywords: 2-edge-coloured graph; alternating hamiltonian cycle; supereulerian; alternating cycle; eulerian factor; extension of a 2-edge-coloured graph.

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“…Suppose W 2 is an even path and W 1 , W 3 are odd paths. Let H i be a graph obtain from H by removing W i for i ∈ [3]. It is obvious that each H i is a 2-connected graph.…”

Section: If N Is Odd Then Whenmentioning

confidence: 99%

“…The three main colored connection colorings: rainbow connection coloring [8], proper connection coloring [5] and proper-walk connection coloring [3], monochromatic connection coloring [6], have been well-studied in recent years. As a counterpart concept of the rainbow connection coloring, rainbow disconnection coloring was introduced in [7] by Chartrand et al in 2018.…”

Section: Introductionmentioning

confidence: 99%

Upper bounds for the $MD$-numbers and characterization of extremal graphs

2020

Preprint

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For an edge-colored graph G, we call an edge-cut M of G monochromatic if the edges of M are colored with the same color. The graph G is called monochromatic disconnected if any two distinct vertices of G are separated by a monochromatic edge-cut. For a connected graph G, the monochromatic disconnection number (or M D-number for short) of G, denoted by md(G), is the maximum number of colors that are allowed in order to make G monochromatic disconnected. For graphs with diameter one, they are complete graphs and so their M D-numbers are 1. For graphs with diameter at least 3, we can construct 2-connected graphs such that their M D-numbers can be arbitrarily large; whereas for graphs G with diameter two, we show that if G is a 2-connected graph then md(G) ≤ 2, and if G has a cut-vertex then md(G) is equal to the number of blocks of G. So, we will focus on studying 2-connected graphs with diameter two, and give two upper bounds of their M D-numbers depending on their connectivity and independent numbers, respectively. We also characterize the n 2 -connected graphs (with large connectivity) whose M D-numbers are 2 and the 2-connected graphs (with small connectivity) whose M D-numbers archive the upper bound n 2 . For graphs with connectivity less than n 2 , we show that if the connectivity of a graph is in linear with its order n, then its M D-number is upper bounded by a constant, and this suggests us to leave a conjecture that for a k-connected graph G, md(G) ≤ n k .

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Proper‐walk connection number of graphs

Abstract: We dedicate this paper to Ron Graham whose enormous energy, friendly attitude, open mindedness and countless contributions to science has always been a great inspiration

Cited by 7 publications

References 20 publications

Graph colorings under global structural conditions

Graph colorings under global structural conditions

Supereulerian 2-edge-coloured graphs

Upper bounds for the $MD$-numbers and characterization of extremal graphs

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