- Tytuł:
- The Vertex-Rainbow Connection Number of Some Graph Operations
- Autorzy:
-
Li, Hengzhe
Ma, Yingbin
Li, Xueliang - Powiązania:
- https://bibliotekanauki.pl/articles/32083892.pdf
- Data publikacji:
- 2021-05-01
- Wydawca:
- Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
- Tematy:
-
rainbow connection number
vertex-rainbow connection number
Cartesian product
lexicographic product
line graph - Opis:
- A path in an edge-colored (respectively vertex-colored) graph G is rainbow (respectively vertex-rainbow) if no two edges (respectively internal vertices) of the path are colored the same. An edge-colored (respectively vertex-colored) graph G is rainbow connected (respectively vertex-rainbow connected) if every two distinct vertices are connected by a rainbow (respectively vertex-rainbow) path. The rainbow connection number rc(G) (respectively vertex-rainbow connection number rvc(G)) of G is the smallest number of colors that are needed in order to make G rainbow connected (respectively vertex-rainbow connected). In this paper, we show that for a connected graph G and any edge e = xy ∈ E(G), rvc(G) ≤ rvc(G − e) ≤ rvc(G) + dG−e(x, y) − 1 if G − e is connected. For any two connected, non-trivial graphs G and H, rad(G□H)−1 ≤ rvc(G□H) ≤ 2rad(G□H), where G□H is the Cartesian product of G and H. For any two non-trivial graphs G and H such that G is connected, rvc(G ◦ H) = 1 if diam(G ◦ H) ≤ 2, rad(G) − 1 ≤ rvc(G ◦ H) ≤ 2rad(G) if diam(G) > 2, where G ◦ H is the lexicographic product of G and H. For the line graph L(G) of a graph G we show that rvc(L(G)) ≤ rc(G), which is the first known nontrivial inequality between the rainbow connection number and vertex-rainbow connection number. Moreover, the bounds reported are tight or tight up to additive constants.
- Źródło:
-
Discussiones Mathematicae Graph Theory; 2021, 41, 2; 513-530
2083-5892 - Pojawia się w:
- Discussiones Mathematicae Graph Theory
- Dostawca treści:
- Biblioteka Nauki
Artykuł