Note on injective edge-coloring of graphs
WebJul 15, 2024 · An injective coloring of a graph is a vertex coloring such that any pair of vertices having a common neighbor receives distinct colors. Panda and Priyamvada [ 12 ] proved complexity results for some subclasses of bipartite graphs, which can be re-interpreted as results on exact square coloring. WebA coloring of edges of a graph G is injective if for any two distinct edges e 1 and e 2, the colors of e 1 and e 2 are distinct if they are at distance 1 in G or in a common triangle. Naturally, the injective chromatic index of G, χ inj (G), is the minimum number of colors needed for an injective edge-coloring of G.
Note on injective edge-coloring of graphs
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WebAug 1, 2024 · The injective chromatic number, χ inj ( G), of a graph G is the minimum k such that G admits injective coloring with k colors. Similarly, an edge coloring of a graph G is … WebMar 31, 2024 · An injective edge-coloring of graph G is an edge coloring φ such that φ (e 1) ≠ φ (e 3) for any three consecutive edges e 1, e 2 and e 3 of a path or a 3-cycle. In other …
WebAn injective edge coloring of a graph G = (V, E) is a coloring c of the edges of G such that if e 1, e 2 and e 3 are consecutive edges in G, then c (e 1) ≠ c (e 3 ). The injective edge coloring number χ i (G) is the minimum number of colors permitted in such a coloring. WebOct 4, 2024 · A k -injective edge-coloring of a graph G is an edge-coloring of G , (not necessarily proper), such that if edges e 1 , e 2 , e 3 are consecutive, then e 1 and e 3 receive distinct colors....
WebMar 1, 2024 · Note that such an edge-coloring is not necessarily proper. The minimum number of colors required for an injective edge-coloring is called the injective chromatic index of G, denoted by χ i ′ ( G ). For every integer k ≥ 2, we show that every k-degenerate graph G with maximum degree Δ satisfies χ i ′ ( G ) ≤ ( 4 k − 3 ) Δ − 2 k 2 − k + 3. WebAn injective edge-coloring c of a graph G is an edge-coloring such that if e 1, e 2, and e 3 are three consecutive edges in G (they are consecutive if they form a path or a cycle of length three), then e 1 and e 3 receive different colors.
WebDec 1, 2024 · An injective k -edge coloring of a graph G= (V (G),E (G)) is a k -edge coloring \varphi of G such that \varphi (e_1)\ne \varphi (e_3) for any three consecutive edges e_1,e_2 and e_3 of a path or a 3-cycle. The injective edge chromatic index of G, denoted by \chi _i' (G), is the minimum k such that G has an injective k -edge coloring.
WebOct 8, 2024 · The central problem of the total-colorings is the Total Coloring Conjecture, which asserts that every graph of maximum degree Δ admits a (Δ+2)-total-coloring. More precisely, this conjecture has been verified for Δ ≤ 5, and it is still open when Δ = 6, even for planar graphs. Let mad ( G) denote the maximum average degree of the graph G. malakoi in the biblemalakoff tx city hallWebAn injective edge-coloring c of a graph G is an edge-coloring such that if e 1, e 2, and e 3 are three consecutive edges in G (they are consecutive if they form a path or a cycle of … malakofftorte thermomixWebMar 1, 2024 · Note that such an edge-coloring is not necessarily proper. The minimum number of colors required for an injective edge-coloring is called the injective chromatic … malakoff tx weather forecasthttp://jeffe.cs.illinois.edu/teaching/comptop/2024/notes/20a-tree-cotree-structures.html malakoff tx isdWebAbstract A k -injective edge coloring of a graph G is a coloring f: E ( G) → C = { 1, 2, 3, …, k }, such that if e 1, e 2 and e 3 are consecutive edges in G, then f ( e 1) ≠ f ( e 3). χ i ′ ( G) = … malak old republic storyWebMay 19, 2024 · The minimum k for which G has a k -injective edge coloring is called the i n j e c t i v e e d g e c o l o r i n g n u m b e r, denoted by χ i ′ ( G). In this paper, we consider … malakpet union bank ifsc code