We prove that for $$n>k\ge 3$$ , if G is an n-vertex graph with chromatic number k but any of its proper subgraphs has smaller number, then contains at most $$n-k+3$$ copies a clique size $$k-1$$ . This answers problem Abbott and Zhou provides tight bound on conjecture Gallai.

A graph G is said to be equimatchable if every matching in G extends to (i.e., is a subset of) a maximum matching. In this paper, we use the Gallai–Edmonds decomposition theory for matchings to determine the equimatchable members of two important graph classes. We 5nd that there are precisely 23 3-connected planar graphs (i.e., 3-polytopes) which are equimatchable and that there are only two cu...

A Gallai coloring is a of the edges complete graph without rainbow triangles, and k-coloring that uses at most k colors. For an integer k≥1, Gallai–Ramsey number GRk(H) given H least positive N such every KN contains monochromatic copy H. Let Cm denote cycle on m≥4 vertices let Θm family graphs obtained from by adding additional edge joining two non-consecutive vertices. We prove GRk(Θ2n+1)=n⋅2...

A d-graph G = (V ; E1, . . . , Ed) is a complete graph whose edges are arbitrarily partitioned into d subsets (colored with d colors); G is a Gallai d-graph if it contains no three-colored triangle ∆; furthermore, G is a CIS d-graph if ⋂d i=1 Si 6= ∅ for every set-family S = {Si | i ∈ [d]}, where Si ⊆ V is a maximal independent set of Gi = (V,Ei), the ith chromatic component of G, for all i ∈ [...

A classical theorem of Gallai states that in every graph that is critical for k-colorings, the vertices of degree k − 1 induce a tree-like graph whose blocks are either complete graphs or cycles of odd length. We provide a generalization to colorings and list colorings of digraphs, where some new phenomena arise. In particular, the problem of list coloring digraphs with the lists at each vertex...

A graph G is called k-critical if it has chromatic number k, but every proper subgraph of it is (k ? 1){colourable. We prove that every k{critical graph (k 6) on n k + 2 vertices has at least 1 2 (k?1+ k?3 (k?c)(k?1)+k?3)n edges where c = (k?5)(1 2 ? 1 (k?1)(k?2)). This improves earlier bounds established by Gallai 9] and, more recently, by Krivelevich 17].

A path cover is a decomposition of the edges of a graph into edge-disjoint simple paths. Gallai conjectured that every connected n-vertex graph has a path cover with at most dn/2e paths. We prove Gallai’s conjecture for series-parallel graphs. For the class of planar 3-trees we show how to construct a path cover with at most b5n/8c paths, which is an improvement over the best previously known b...

We study the eigenspace with largest eigenvalue of the Laplacian matrix of a simple graph. We find a surprising connection of this space with the theory of modular decomposition of Gallai, whereby eigenvectors can be used to discover modules. In the case of comparability graphs, eigenvectors are used to induce orientations of the graph, and the set of these induced orientations is shown to (rec...

We study the eigenspace with largest eigenvalue of the Laplacian matrix of a simple graph. We find a surprising connection of this space with the theory of modular decomposition of Gallai, whereby eigenvectors can be used to discover modules. In the case of comparability graphs, eigenvectors are used to induce orientations of the graph, and the set of these induced orientations is shown to (rec...

We show some consequences of results of Gallai concerning edge colorings of complete graphs that contain no tricolored triangles. We prove two conjectures of Bialostocki and Voxman about the existence of special monochromatic spanning trees in such colorings. We also determine the size of largest monochromatic stars guaranteed to occur. 2004 Wiley Periodicals, Inc. J Graph Theory 46: 211–216, 2004