Note on injective edge-coloring of graphs
WebJul 23, 2024 · An injective edge-coloring of a graph is an edge-coloring such that if , , and are three consecutive edges in (they are consecutive if they form a path or a cycle of length three), then and receive different colors. The minimum integer such that, has an injective edge-coloring with colors, is called the injective chromatic index of ( ). 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.
Note on injective edge-coloring of graphs
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WebAn injective edge-coloring c of a graph G is an edge-coloring such that if e1, e2, and e3 are three consecutive edges in G (they are consecutive if they form a path or a cycle of length …
WebAn injective k -edge coloring of a graph G = ( V ( G), E ( G)) is a k -edge coloring φ such that if e 1 and e 2 are at distance exactly 2 or in the same triangle, then φ ( e 1) ≠ φ ( e 2). The injective chromatic index of G, denoted by χ i ′ ( G), is the minimum k such that G has an injective k -edge coloring. The edge weight of G is ... 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.
WebMay 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 … WebAbstract 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) = …
WebFeb 2, 2024 · An injective edge coloring of a graph G = (V, E) is a coloring c of the edges of G such that if e1,e2 and e3 are consecutive edges in G, then c (e1) c (e3). The injective …
WebEquivalently, suppose we color the edges of \(\Sigma\) red, green, or blue in arbitrary order so that (1) every cycle contains at least one blue edge, (2) every edge cut contains at least one red edge, and (3) an edge is green if and only if it cannot be colored either red or blue. Then the red, green, and blue edges respectively define the ... csi: cyber the evil twin imdbWebAn embedding ˙of a graph G= (V;E) in a surface is an injective ... coloring ’of a graph Gis a mapping from V(G) to a set of colors Csuch that ’(u) 6= ’(v) whenever uv2E(G). ... Theorem 2 ([2, 17]). Let Gbe a triangle-free planar graph and Hbe a graph such that G= H hfor some edge hof H. Then His 3-colorable. Theorem 3 ([17]). Let Gbe a ... eagle co sylvan lake fireWebAn 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 … csi dark motives windows 10 patchWebMar 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. csi: cyber why-fi imdbWebMay 19, 2024 · In this paper, we consider the injective edge coloring numbers of generalized Petersen graphs P ( n, 1) and P ( n, 2). We determine the exact values of injective edge coloring numbers for P ( n, 1) with n ≥ 3, and for P ( n, 2) with 4 ≤ n ≤ 7. For n ≥ 8, we show that 4 ≤ χ i ′ ( P ( n, 2)) ≤ 5. Keywords: k -injective edge coloring, csi cyber shades of grayWebA 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) = min { k G has a k -injective edge coloring } is called the injective edge coloring number. csi cyber showWebOct 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.... csi cyber shades of grey