Albertson conjectured that if graph G has chromatic number r, then the crossing number of G is at least that of the complete graph Kr. This conjecture in the case r = 5 is equivalent to the four color theorem. It was verified for r = 6 by Oporowski and Zhao. In this paper, we prove the conjecture for 7 ≤ r ≤ 12 using results of Dirac; Gallai; and Kostochka and Stiebitz that give lower bounds on...

Erd os and Gallai showed that for any simple graph with n vertices and circumference c it holds that |E(G)| ≤ 2 (n−1)c.We extend this theorem to simple binary matroids having no F7-minor by showing that for such a matroid M with circumference c(M)≥ 3 it holds that |E(M)| ≤ 1 2 r(M)c(M). Mathematics Subject Classi cation (2000). Primary 05D15; Secondary 05B35.

Tibor Gallai conjectured that the edge set of every connected graph G on n vertices can be partitioned into ⌈n/2⌉ paths. Let Gk be the class of all 2k-regular graphs of girth at least 2k − 2 that admit a pair of disjoint perfect matchings. In this work, we show that Gallai’s conjecture holds in Gk, for every k ≥ 3. Further, we prove that for every graph G in Gk on n vertices, there exists a par...

Given two non-empty graphs G, H and a positive integer k, the Gallai–Ramsey number $${\text {gr}}_k(G:H)$$ is defined as minimum N such that for all $$n\ge N$$ , every k-edge-coloring of $$K_n$$ contains either rainbow colored copy G or monochromatic H. In this paper, we got some exact values bounds {gr}}_k(P_5:H) \ (k\ge 3)$$ if general graph star with extra independent edges pineapple.

We give asymptotically sharp upper bounds for the maximum diameter and radius of (i) a connected graph, {ii) a connected trangle-free graph, (iii) a connected C,-free graph with n vertices and with minimum degree 6, where n tends to infinity. Some conjectures for J&-free graphs are also stated. Let G be a connected graph with vertex set V(G) and edge set E(G). For any XJJ E V(G) let dG(x,y) den...

This paper is on matching theory. A graph with perfect matching is called saturated if any addition of one complement edge creates a new perfect matching. Lovász gave a characterization of the saturated graphs called the cathedral theorem, and later Szigeti gave another proof. In this paper, we gave a new proof with our preceding works, which revealed the canonical structure of general graph wi...

A problem, first considered by Erdős and Gallai [2], was to determine the maximum number of edges in any graph on n vertices if it contains no path with k + 1 vertices. This maximum number, ext(n, Pk+1), is called the extremal number for the path Pk+1. Erdős and Gallai proved the following theorem, which was one of the earliest extremal results in graph theory. Theorem 1.1 ([2]). For every k ≥ ...

Erdős and Gallai proved that a nonincreasing list (d1, . . . , dn) of nonnegative integers is the list of degrees of a graph (with no loops or multi-edges) if and only if the sum is even and the list satisfies ∑k i=1 di ≤ k(k−1)+ ∑n i=k+1 min{k, di} for 1 ≤ k ≤ n. We give a short constructive proof of the characterization. AMS Subject classification: 05C07

Given a graph G and positive integer k, define the Gallai—Ramsey number to be minimum of vertices n such that any k-edge coloring Kn contains either rainbow (all different colored) triangle or monochromatic copy G. In this paper, we obtain exact values numbers for union two stars in many cases bounds other cases. This work represents first class disconnected graphs considered as desired subgraph.

Given a graph G and positive integer k, define the Gallai-Ramsey number to be minimum of vertices n such that any k-edge coloring $$K_n$$ contains either rainbow (all different colored) triangle or monochromatic copy G. Much like Ramsey numbers, numbers have gained reputation as being very difficult compute in general. As yet, still only precious few sharp results are known. In this paper, we o...