3-star factors in random d-regular graphs

AbstractThe small subgraph conditioning method first appeared when Robinson and the second author showed the almost sure hamiltonicity of random d-regular graphs. Since then it has been used to study the almost sure existence of, and the asymptotic distribution of, regular spanning subgraphs of various types in random d-regular graphs and hypergraphs. In this paper, we use the method to prove the almost sure existence of 3-star factors in random d-regular graphs. This is essentially the first application of the method to non-regular subgraphs in such graphs

On The Total Irregularity Strength of Regular Graphs

Let ðº = (ð‘‰, ð¸) be a graph. A total labeling ð‘“: 𑉠∪ ð¸ → {1, 2, ⋯ , ð‘˜} iscalled a totally irregular total ð‘˜-labeling of ðº if every two distinct vertices ð‘¥ and𑦠in 𑉠satisfy ð‘¤ð‘“(ð‘¥) ≠ ð‘¤ð‘“(ð‘¦) and every two distinct edges ð‘¥1ð‘¥2 and ð‘¦1ð‘¦2 in ð¸satisfy ð‘¤ð‘“(ð‘¥1ð‘¥2) ≠ ð‘¤ð‘“(ð‘¦1ð‘¦2), where ð‘¤ð‘“(ð‘¥) = ð‘“(ð‘¥) + Σð‘¥ð‘§âˆˆð¸(ðº) ð‘“(ð‘¥ð‘§) andð‘¤ð‘“(ð‘¥1ð‘¥2) = ð‘“(ð‘¥1) + ð‘“(ð‘¥1ð‘¥2) + ð‘“(ð‘¥2). The minimum 𑘠for which a graph ðº hasa totally irregular total ð‘˜-labeling is called the total irregularity strength of ðº,denoted by ð‘¡ð‘ (ðº). In this paper, we consider an upper bound on the totalirregularity strength of ð‘š copies of a regular graph. Besides that, we give a dual labeling of a totally irregular total ð‘˜-labeling of a regular graph and we consider the total irregularity strength of ð‘š copies of a path on two vertices, ð‘š copies of a cycle, and ð‘š copies of a prism ð¶ð‘› â–¡ ð‘ƒ2

BEBERAPA KELAS GRAF RAMSEY MINIMAL UNTUK LINTASAN P_3 VERSUS P_5

In 1930, Frank Plumpton Ramsey has introduced Ramsey's theory, in his paper titled On a Problem of Formal Logic. This study became morepopular since Erdős and Szekeres applied Ramsey's theory to graph theory. Suppose given the graph F, G and H. The notation F → (G, H)  states thatfor any red-blue coloring of the edges of F implies F containing a red subgraph of G or a blue subgraph of H. The graph F is said to be the Ramsey graph for graph G versus H (pair G and H) if F → (G, H). Graph F is called Ramsey minimal graph for G versus H if  first, F → (G, H) and second, F satisfies the minimality property i.e. for each e ∈ E (F), then F-e ↛ (G, H). The class of all Ramsey (G, H) minimal graphs is denoted by (G, H). The class (G, H) is called Ramsey infinite or finite if  (G, H) is infinite or finite, respectively. The study about Ramsey minimal graph is still continuously being developed and examined, although in general it is not easy to characterize or determine the graphs included in the (G, H), especially if  (G, H) is an infinite Ramsey class. The characterization of graphs in (, ) has been obtained. However, the characterization of graphs in (, ), for every 3 ≤ m < n is still open. In this article, we will determine some infinite classes of Ramsey minimal graphs  for paths  versus .

Trees with Certain Locating-chromatic Number

The locating-chromatic number of a graph G can be defined as the cardinality of a minimum resolving partition of the vertex set V(G) such that all vertices have distinct coordinates with respect to this partition and every two adjacent vertices in G are not contained in the same partition class. In this case, the coordinate of a vertex v in G is expressed in terms of the distances of v to all partition classes. This concept is a special case of the graph partition dimension notion. Previous authors have characterized all graphs of order n with locating-chromatic number either n or n-1. They also proved that there exists a tree of order n, n≥5, having locating-chromatic number k if and only if k âˆˆ{3,4,"¦,n-2,n}. In this paper, we characterize all trees of order n with locating-chromatic number n - t, for any integers n and t, where n &gt; t+3 and 2 ≤ t &lt; n/2

The Ramsey numbers for disjoint unions of trees

For given graphs G and H, the Ramsey number R(G,H) is the smallest natural number n such that for every graph F of order\ud 9 n: either F contains G or the complement of F contains H. In this paper, we investigate the Ramsey number R(???G,H), where G\ud is a tree and H is a wheel Wm or a complete graph Km. We show that if n 3, then R(kSn,W4) = (k + 1)n for k 2, even n and\ud R(kSn,W4) = (k + 1)n ??? 1 for k 1 and odd n.We also show that R(\ud \ud k\ud i=1Tni,Km) = R(Tnk,Km) +\ud \ud k???1\ud 11 i=1 ni

Star-Whell Ramsey Numbers

For given graphs G and H; the Ramsey number R(G;H)\ud is the smallest natural number n such that for every graph F of order\ud n: either F contains G or the complement of F contains H: This paper\ud investigates the Ramsey number R(Sn;Wm) of stars versus wheels, where\ud n is smaller than or equal to m. We show that if m is odd and n + 1 ??\ud m ?? 2n ?? 4, then R(Sn;Wm) = 3n ?? 2: Furthermore, if n is odd, n ?? 5\ud and m > n, then R(Sn;Wm) = 3n ?? ??, where ?? = 4 if m = 2n ?? 4 and\ud ?? = 6 if m = 2n ?? 8 or m = 2n ?? 6.\ud

The Ramsey numbers for disjoint unions of graphs

Abstract. For given graphs G and H, the Ramsey number R(G,H)\ud is the smallest natural number n such that for every graph F of order\ud n: either F contains G or the complement of F contains H. This paper\ud investigates the Ramsey number R(kG,H) for any natural number k.We\ud show that if 2n???4, 2n???8 or 2n???6, then R(kSn,Wm) = R(Sn,Wm)+\ud (k???1)n. Furthermore, if |Gi| (|Gi|???|Gi+1|)( (H)???1) and R(Gi,H) =\ud ( (H)???1)(|Gi|???1)+1, , for each i, then R(\ud Sk\ud P i=1 Gi,H) = R(Gk,H)+ k???1\ud i=1 |Gi|

Small Maximal Matchings of Random Cubic Graphs

We consider the expected size of a smallest maximalmatching of cubic graphs. Firstly, we present a randomized greedy algorithm for finding a small maximal matching of cubic graphs.We analyze the averagecase performance of this heuristic on random n-vertex cubic graphs using differential equations. In this way, we prove that the expected size of the maximalmatching returned by the algorithm is asymptotically almost surely (a.a.s.) less than 0.34623n. We also give an existence proof which shows that the size of a smallest maximal matching of a random n-vertex cubic graph is a.a.s. less than 0.3214n. It is known that the size of a smallest maximal matching of a random n-vertex cubic graph is a.a.s. larger than 0.3158n

MATERI PENGAYAAN TEORI BILANGAN DASAR DI SEKOLAH DASAR

Kemampuan matematika siswa di Indonesia mengalami penurunan, hal ini didasari hasil PISA Indonesia yang mengalami penurunan dari sebelumnya memiliki skor 386 menjadi 379 [2]. Zaskiz [5] berpendapat bahwa terdapat celah antara hubungan aritmatika dan aljabar, sehingga banyak siswa yang kesulitan ketika transisi dari Sekolah Dasar ke Sekolah Menengah Pertama. Campbell [3] mengklaim bahwa teori bilangan dapat menjembatani celah yang ada antara aritmatika dan aljabar. Miele [4] menyimpulkan teori bilangan dapat meningkatkan kemampuan metakognitif dan sikap siswa terhadap matematika. Penyebab lain dari turunnya kemampuan matematika siswa Indonesia adalah cara penyampaian pembelajaran yang masih konvensional. Hal ini mengakibatkan siswa cenderung tidak belajar dan hanya menerima apa yang diberikan oleh gurunya, sehingga pengetahuan yang didapat siswa menjadi kurang bermakna. Meskipun begitu beragamnya pembelajaran matematika yang diajarkan di berbagai belahan dunia, tetapi kehadiran teori bilangan dalam kurikulum sedikit sekali [5]. Pada Kurikulum 2013 konsep teori bilangan di Sekolah Dasar maupun di Sekolah Menengah Pertama hanyalah faktorisasi prima, faktor persekutuan terbesar (fpb), dan kelipatan persekutuan terkecil (kpk). Oleh karenanya pada makalah ini, kami membuat rancangan pembelajaran dan materi ajar pengayaan mengenai teori bilangan dasar, yaitu kekongruenan menggunakan metode pembelajaran Discovery Learning. Rancangan pembelajaran ini diharapkan dapat meningkatkan kemampuan matematika dan sikap siswa di Sekolah Dasar