Inspired by recent algorithms for electing a leader in a distributed system, we study the following game in a directed graph: each vertex selects one of its outgoing arcs (if any) and eliminates the other endpoint of this arc; the remaining vertices play on until no arcs remain. We call a directed graph lethal if the game must end with all vertices eliminated and mortal if it is possible that t...

A colored graph is a complete graph in which a color has been assigned to each edge, and a colorful cycle is a cycle in which each edge has a different color. We first show that a colored graph lacks colorful cycles iff it is Gallai, i.e., lacks colorful triangles. We then show that, under the operation m◦n ≡ m+n−2, the omitted lengths of colorful cycles in a colored graph form a monoid isomorp...

Putting the concept of line graph in a more general setting, for a positive integer k the k-line graph Lk(G) of a graph G has the Kk-subgraphs of G as its vertices, and two vertices of Lk(G) are adjacent if the corresponding copies of Kk in G share k− 1 vertices. Then, 2-line graph is just the line graph in usual sense, whilst 3-line graph is also known as triangle graph. The k-anti-Gallai grap...

We make progress on three long standing conjectures from the 1960s about path and cycle decompositions of graphs. Gallai conjectured that any connected graph n vertices can be decomposed into at most ? 2 ? paths, while a conjecture Hajós states Eulerian ? ? 1 ? cycles. The Erd?s–Gallai O ( ) cycles edges. show if G is sufficiently large with linear minimum degree, then following hold. (i) + o p...

Gallai conjectured that every 4-critical graph on n vertices has at least 53n − 23 edges. We prove this conjecture for 4-critical graphs in which the subgraph induced by vertices of degree 3 is connected.

We show that given any vertex x of a k-colour-critical graph G with a connected complement, the graph G − x can be (k − 1)-coloured so that every colour class contains at least 2 vertices. This extends the well-known theorem of Gallai, that a k-colour-critical graph with a connected complement has at least 2k − 1 vertices. Our proof does not use matching theory. It is considerably shorter, conc...

In “A note on a theorem of Erdös & Gallai” ([6]) one identifies the nonredundant inequalities in a characterization of graphical sequences. We explain how this result may be obtained directly from a simple geometrical observation involving weak majorization. A sequence of positive integers d1, d2, . . . , dp is called graphical if it is the degree sequence of a graph, i.e., there is a graph who...

In the 1960s, Erdős and Gallai conjectured that the edge set of every graph on n vertices can be partitioned into O(n) cycles and edges. They observed that one can easily get an O(n log n) upper bound by repeatedly removing the edges of the longest cycle. We make the first progress on this problem, showing that O(n log log n) cycles and edges suffice. We also prove the Erdős-Gallai conjecture f...

In this paper, two notions, the clique irreducibility and clique vertex irreducibility are discussed. A graph G is clique irreducible if every clique in G of size at least two, has an edge which does not lie in any other clique of G and it is clique vertex irreducible if every clique in G has a vertex which does not lie in any other clique of G. It is proved that L(G) is clique irreducible if a...

It is proven that for k ≥ 4 and n > k every k-color-critical graph on n vertices has at least ( k−1 2 + k−3 2(k2−2k−1) ) n edges, thus improving a result of Gallai from 1963. A graph G is k-color-critical (or simply k-critical) if χ(G) = k but χ(G′) < k for every proper subgraph G′ of G, where χ(G) denotes the chromatic number of G. (See, e.g., [2] for a detailed account of graph coloring probl...