The structure of edge-colored complete graphs containing no properly colored triangles has been characterized by Gallai back in the 1960s. More recently, Cǎda et al. and Fujita independently determined bipartite C 4 . We characterize even cycles, or equivalently, without a 6 In particular, we first deal with simple case 2-edge-colored graphs, using result Yeo. Next, for k ≥ 3, define four class...

A graph G is k-critical if G is not (k − 1)-colorable, but every proper subgraph of G is (k − 1)-colorable. A graph G is k-choosable if G has an L-coloring from every list assignment L with |L(v)| = k for all v, and a graph G is k-list-critical if G is not (k − 1)-choosable, but every proper subgraph of G is (k − 1)-choosable. The problem of bounding (from below) the number of edges in a k-crit...

Given a graph G, its triangular line graph is the graph T (G) with vertex set consisting of the edges of G and adjacencies between edges that are incident in G as well as being within a common triangle. Graphs with a representation as the triangular line graph of some graph G are triangular line graphs, which have been studied under many names including anti-Gallai graphs, 2-in-3 graphs, and li...

For a graph $H$ and an integer $k\ge1$, the $k$-color Ramsey number $R_k(H)$ is least $N$ such that every $k$-coloring of edges complete $K_N$ contains monochromatic copy $H$. Let $C_m$ denote cycle on $m\ge4$ vertices let $\Theta_m$ family graphs obtained from by adding additional edge joining two non-consecutive vertices. Unlike odd cycles, little known about general behavior $R_k(C_{2n})$ ex...

We show that two results on covering of edge colored graphs by monochromatic connected parts can be extended to partitioning. We prove that for any 2-edgecolored non-trivial r-uniform hypergraph H, the vertex set can be partitioned into at most α(H)− r+ 2 monochromatic connected parts, where α(H) is the maximum size of a set of vertices that does not contain any edge. In particular, any 2-edgec...

A graph G is called k-critical if x(G) = k and x(G e) -C x(G) for each edge e of G, where x denotes the chromatic number. T. Gallai conjectured that every k-critical graph of order n contains at most n complete (k l)-subgraphs. In 1987, Stiebitz proved Gallai’s conjecture in the case k = 4, and in 1992 Abbott and Zhou proved Gallai’s conjecture for all k > 5. In their paper, Abbott and Zhou ask...

We present exact solutions for the size of giant connected component (GCC) graphs composed higher-order homogeneous cycles, including weak cycles and cliques, following bond percolation. use our theoretical result to find location percolation threshold model, providing analytical where possible. expect results derived here be useful a wide variety applications graph theory, epidemiology, lattic...

Gallai-colorings of complete graphs—edge colorings such that no triangle is colored with three distinct colors—occur in various contexts such as the theory of partially ordered sets (in Gallai’s original paper), information theory and the theory of perfect graphs. A basic property of GallaiQ1 colorings with at least three colors is that at least one of the color classes must span a disconnected...

We present a short proof of the Berge–Tutte Formula and the Gallai–Edmonds Structure Theorem from Hall’s Theorem. The fundamental theorems on matchings in graphs have been proved in many ways. The most famous of these results is Hall’s Theorem [6], characterizing when a bipartite graph has a matching that covers one partite set. Anderson [1] used Hall’s Theorem to prove Tutte’s 1Factor Theorem ...

In 1966 Gallai asked whether all longest paths in a connected graph have nonempty intersection. This is not true in general and various counterexamples have been found. However, the answer to Gallai’s question is positive for several well-known classes of graphs, as for instance connected outerplanar graphs, connected split graphs, and 2-trees. A graph is series-parallel if it does not contain ...