With the aim of building a more precise mathematical model of better predictability for the formation of the supernumerary rainbow and fogbow and seeking a clearer and more elaborate physical interpretation, this pape...With the aim of building a more precise mathematical model of better predictability for the formation of the supernumerary rainbow and fogbow and seeking a clearer and more elaborate physical interpretation, this paper examines the relationship between different rainbow patterns and droplet sizes through both analytical derivation and numerical simulation and develops a much more detailed model beyond previous explanations. From Newton’s geometric model of optics to Young’s wave model, the paper first establishes a solid foundation for the understanding of the formation of the rainbow in nature and through human vision, and then goes on examining the interferences of light, finally applying the model in reality for a better understanding of complex rainbow phenomena, with additional analysis on an unexpected finding about the correspondence of maximum view angle and shortest light path through hypothetical explanation based on the principle of least time and simulation of an elliptical droplet.展开更多
Abstract A total-colored path is total rainbow if its edges and internal vertices have distinct colors. A total-colored graph G is total rainbow connected if any two distinct vertices are connected by some total rainb...Abstract A total-colored path is total rainbow if its edges and internal vertices have distinct colors. A total-colored graph G is total rainbow connected if any two distinct vertices are connected by some total rainbow path. The total rainbow connection number of G, denoted by trc（G）, is the smallest number of colors required to color the edges and vertices of G in order to make G total rainbow connected. In this paper, we investigate graphs with small total rainbow connection number. First, for a connected graph G, we prove that trc（G） = 3 if （n-12） ＋ 1 ≤ ｜E（G）｜≤ （n2） - 1, and trc（G） ≤ 6 if ｜E（G）｜≥ （n22） ＋2. Next, we investigate the total rainbow connection numbers of graphs G with ｜V（G）｜ = n, diam（G） ≥ 2, and clique number w（G） = n - s for 1 ≤ s ≤ 3. In this paper, we find Theorem 3 of [Discuss. Math. Graph Theory, 2011, 31（2）： 313-320] is not completely correct, and we provide a complete result for this theorem.展开更多
Given two non-empty graphs G,H and a positive integer k,the Gallai-Ramsey number gr_(k)(G:H)is defined as the minimum integer N such that for all n≥N,every exact k-edge-coloring of Kn contains either a rainbow copy o...Given two non-empty graphs G,H and a positive integer k,the Gallai-Ramsey number gr_(k)(G:H)is defined as the minimum integer N such that for all n≥N,every exact k-edge-coloring of Kn contains either a rainbow copy of G or a monochromatic copy of H.Denote gr′_(k)(G:H)as the minimum integer N such that for all n≥N,every edge-coloring of Kn using at most k colors contains either a rainbow copy of G or a monochromatic copy of H.In this paper,we get some exact values or bounds for gr_(k)(P_(5):H)and gr′_(k)(P_(5):H),where H is a cycle or a book graph.In addition,our results support a conjecture of Li,Besse,Magnant,Wang and Watts in 2020.展开更多
文摘With the aim of building a more precise mathematical model of better predictability for the formation of the supernumerary rainbow and fogbow and seeking a clearer and more elaborate physical interpretation, this paper examines the relationship between different rainbow patterns and droplet sizes through both analytical derivation and numerical simulation and develops a much more detailed model beyond previous explanations. From Newton’s geometric model of optics to Young’s wave model, the paper first establishes a solid foundation for the understanding of the formation of the rainbow in nature and through human vision, and then goes on examining the interferences of light, finally applying the model in reality for a better understanding of complex rainbow phenomena, with additional analysis on an unexpected finding about the correspondence of maximum view angle and shortest light path through hypothetical explanation based on the principle of least time and simulation of an elliptical droplet.
文摘Abstract A total-colored path is total rainbow if its edges and internal vertices have distinct colors. A total-colored graph G is total rainbow connected if any two distinct vertices are connected by some total rainbow path. The total rainbow connection number of G, denoted by trc（G）, is the smallest number of colors required to color the edges and vertices of G in order to make G total rainbow connected. In this paper, we investigate graphs with small total rainbow connection number. First, for a connected graph G, we prove that trc（G） = 3 if （n-12） ＋ 1 ≤ ｜E（G）｜≤ （n2） - 1, and trc（G） ≤ 6 if ｜E（G）｜≥ （n22） ＋2. Next, we investigate the total rainbow connection numbers of graphs G with ｜V（G）｜ = n, diam（G） ≥ 2, and clique number w（G） = n - s for 1 ≤ s ≤ 3. In this paper, we find Theorem 3 of [Discuss. Math. Graph Theory, 2011, 31（2）： 313-320] is not completely correct, and we provide a complete result for this theorem.
基金supported by the Shanghai Sailing Program(No.22YF1416000)the National Natural Science Foundation of China(Nos.12401461,12471329,12061059,12201395).
文摘Given two non-empty graphs G,H and a positive integer k,the Gallai-Ramsey number gr_(k)(G:H)is defined as the minimum integer N such that for all n≥N,every exact k-edge-coloring of Kn contains either a rainbow copy of G or a monochromatic copy of H.Denote gr′_(k)(G:H)as the minimum integer N such that for all n≥N,every edge-coloring of Kn using at most k colors contains either a rainbow copy of G or a monochromatic copy of H.In this paper,we get some exact values or bounds for gr_(k)(P_(5):H)and gr′_(k)(P_(5):H),where H is a cycle or a book graph.In addition,our results support a conjecture of Li,Besse,Magnant,Wang and Watts in 2020.
