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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2403.02826 |
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| _version_ | 1866910409322135552 |
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| author | Mirdamad, Shahrzad Sadat Mojdeh, Doost Ali |
| author_facet | Mirdamad, Shahrzad Sadat Mojdeh, Doost Ali |
| contents | An injective coloring of a given graph G = (V, E) is a vertex coloring of G such that any two vertices with common neighbor receive distinct colors. An e-injective coloring of a graph G is a vertex coloring of G such that any two vertices with common edge neighbor receive distinct colors; in the other words, if u and v are the end of a path P4 in a graph G, then they are assigned with different labels. With this new definition, we want to take a review at injective coloring of a graph from the new point of view. For this purpose, we will have a comparison between e-injective coloring with usual coloring, injective coloring, and 2-distance coloring. As well, we review the conjectures raised so far in the literature of injective coloring and 2-distance coloring, from the new approach, e-injective coloring. Finally, we precisely investigate the e-injective coloring of trees, join of two graphs, a family of standard graphs, grid graphs, cylinder graphs and tori graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_02826 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | e-injective coloring: injective and 2-distance colorings conjectures Mirdamad, Shahrzad Sadat Mojdeh, Doost Ali Combinatorics An injective coloring of a given graph G = (V, E) is a vertex coloring of G such that any two vertices with common neighbor receive distinct colors. An e-injective coloring of a graph G is a vertex coloring of G such that any two vertices with common edge neighbor receive distinct colors; in the other words, if u and v are the end of a path P4 in a graph G, then they are assigned with different labels. With this new definition, we want to take a review at injective coloring of a graph from the new point of view. For this purpose, we will have a comparison between e-injective coloring with usual coloring, injective coloring, and 2-distance coloring. As well, we review the conjectures raised so far in the literature of injective coloring and 2-distance coloring, from the new approach, e-injective coloring. Finally, we precisely investigate the e-injective coloring of trees, join of two graphs, a family of standard graphs, grid graphs, cylinder graphs and tori graphs. |
| title | e-injective coloring: injective and 2-distance colorings conjectures |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2403.02826 |