Consider the family of all finite graphs with maximum degree ∆(G) < d and matching number ν(G) < m. In this paper we give a new proof to obtain the exact upper bound for the number of edges in such graphs and also characterize all the cases when the maximal graph is unique. We also provide a new proof of Gallai’s lemma concerning factor critical graphs.

We determine the 2-color Ramsey number of a connected triangle matching c(nK3) which is a graph with n vertex disjoint triangles plus (at least n − 1) further edges that keep these triangles connected. We obtain that R(c(nK3), c(nK3)) = 7n − 2, somewhat larger than in the classical result of Burr, Erdős and Spencer for a triangle matching, R(nK3, nK3) = 5n. The motivation is to determine the Ra...

We show that if G is a bipartite graph with no induced cycles on exactly 6 vertices, then the minimum number of chain subgraphs of G needed to cover E(G) equals the chromatic number of the complement of the square of line graph of G. Using this, we establish that for a chordal bipartite graph G, the minimum number of chain subgraphs of G needed to cover E(G) equals the size of a largest induced...

We give a formula for the v-number of graded ideal that can be used to compute this number. Then, we show edge I(G) graph G, induced matching number G is an upper bound when very well-covered, or has simplicial partition, well-covered connected and contains neither four, nor five cycles. In all these cases, lower regularity ring G. classify cycle vertices are shedding gain insight into family W...

The induced matching partition number of a graph G, denoted by imp(G), is the minimum integer k such that V(G) has a k-partition (V1, V2 ... Vk ) such that, for each i, 1 ≤ i ≤ k, G[Vi], the subgraph of G induced by Vi, is a 1-regular graph. The induced matching k-partition problem is NP-complete even for k = 2. In this paper we investigate the induced matching partition problem for butterfly n...

Erdős and Sós proposed a problem of determining the maximum number F (n) of rainbow triangles in 3-edge-colored complete graphs on n vertices. They conjectured that F (n) = F (a)+ F (b) +F (c) +F (d) + abc+ abd+ acd+ bcd, where a+ b+ c+ d = n and a, b, c, d are as equal as possible. We prove that the conjectured recurrence holds for sufficiently large n. We also prove the conjecture for n = 4 f...

We study the number of matching substrings in the pattern matching problem. In general, there can be a quadratic number of matching substrings in the size of a given text. The linearizing restriction enables to find at most a linear number of matching substrings. We first explore two well-known linearizing restriction rules, the longest-match rule and the shortest-match substring search rule, a...

As an application of CCTV Traffic surveillance, retrieval of the number plate from the vehicles is an important dimension, which demands intelligent solution. In this document, template matching block of computer vision toolbox has been used to extract the vehicle number plate. It is assumed that images of the vehicle have been captured at a particular resolution and orientation. It is also ass...

An r-edge coloring of a graph G is a mapping h : E(G) → [r], where h(e) is the color assigned to edge e ∈ E(G). An exact r-edge coloring is an r-edge coloring h such that there exists an e ∈ E(G) with h(e) = i for all i ∈ [r]. Let h be an edge coloring of G. We say G is rainbow if no two edges in G are assigned the same color by h. The anti-Ramsey number, AR(G,n), is the smallest integer r such...