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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WebOct 27, 2024 · Injective chromatic index is closely related to strong edge-coloring. A proper injective edge-coloring is exactly a strong edge-coloring, which partitions the edges of a … 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) = …
WebMay 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,
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 … WebJan 7, 2024 · Note that an injective edge coloring is not necessarily a proper edge coloring. The notion of injective edge coloring was introduced in 2015 by Cardoso et al. ( 2024) …
WebOct 20, 2016 · An injective edge coloring of a graph G is an edge coloring of G such that if e1, e2 and e3 are consecutive edges in G, then e1 and e3 receive the different colors. The injective edge coloring number or injective edge chromatic index of a graph G, denoted by χ i ′ ( G), is the minimum number of colors permitted in an injective edge coloring of G.
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 ... slow fashion mouvementWebAn 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 … slowfashionnextWebAn 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 ... software for cricut expressionWebA 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. slowfashionnext.comWebJul 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 ( ). software for cricut joyWebAn 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. software for credit analysisWebDec 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. software for cricut explorer air 2