Maximal k-Edge-Colorable Subgraphs, Vizing's Theorem, and Tuza's Conjecture · arXivDesk
1510.07017Oct 23, 201511 pages, 1 figure. Fixed some inaccurate references to "Vizing's Theorem" (the stronger version cited here is in fact due to Ore), cleared up some muddled results in the section about forests, simplified some notation, and made other various readability improvements
Maximal k-Edge-Colorable Subgraphs, Vizing's Theorem, and Tuza's Conjecture
We prove that if M is a maximal k-edge-colorable subgraph of a multigraph G and if F={v∈V(G):dM(v)≤k−μ(v)}
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, then
dF(v)≤dM(v)
for all
v∈F
. (When
G
is a simple graph, the set
F
is just the set of vertices having degree less than
k
in
M
.) This implies Vizing's Theorem as well as a special case of Tuza's Conjecture on packing and covering of triangles. A more detailed version of our result also implies Vizing's Adjacency Lemma for simple graphs.