We prove the 3-Decomposition Conjecture: every finite connected cubic loopless multigraph decomposes into a spanning tree, a 2-regular subgraph, and a matching. The proof rests on a new theorem on matching complements in subcubic graphs. Let H be a finite connected bridgeless simple graph of maximum degree three, and let S be its set of degree-two vertices, with |S| = k>= 2. We show that H has a matching of size equal to its cyclomatic number, |E(H)| - |V(H)| + 1, whose deletion leaves a single tree containing all of S, together with cycles disjoint from S. The proof is by induction on k, using an alternating-path exchange that stops at the first entry into the growing tree component.
We prove that the realization graph of every graphical degree sequence is maximally Hamiltonian: it is Hamilton-laceable when bipartite on more than one vertex, and Hamilton-connected otherwise. This answers Problem P59 of M\"utze's survey of combinatorial Gray codes, and the Hamiltonicity question recorded as open by...
An acyclic edge coloring of a graph G is a proper edge coloring such that G contains no bichromatic cycles. The acyclic chromatic index χa′(G) is the minimum number of colors required for an acyclic edge coloring. Fiamčik and Alon et al. independently conjectured that χa′(G)≤Δ+2 for every simple graph G with maximum de...
A prime labeling of a graph labels its vertices bijectively with the integers from one to the number of vertices, so that adjacent vertices receive coprime labels. I prove the Entringer-Tout conjecture that every tree admits a prime labeling, extending the result of Haxell, Pikhurko, and Taraz from sufficiently large t...
We prove three theorems on the flow monoids of finite graphs. First, we show that a non-empty finite graph G = (V, E) is connected if and only if its flow monoid contains a constant map on V, equivalently, if and only if it contains all constant maps on V. Second, we give a new characterization of graph minors in terms...
A. Assem, H. Derets, C. Nehaniv· Electronic Proceedings in Th...· 0 citations
A graph is
perfect
if, for every induced subgraph, the chromatic number equals the size of its largest clique. In 1972, Lovász established a fundamental characterization of perfect graphs, showing that a graph is perfect if and only if, for every induced subgraph, the product of the size of the largest independen...
A graph is integral if the spectrum of its adjacency matrix consists entirely of integers. We prove that every simple graph having a pendant path with at least three edges has an eigenvalue in $(1,2\cos(\pi/9)]$ and one in $[-2\cos(\pi/9),-1)$, and hence is not integral. This settles a conjecture of Braga, Del-Vecchio...
R. O. Braga, Jean Carlo Moraes, Matheus C. Santos· 1 citation
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