Geodesic
A characterization of 4-χ ρ -(vertex-)critical graphs
Filomat, Tome 36 (2022) no. 19, p. 6481

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DOI

Given a graph G, a function c : V(G) −→ {1, . . . , k} with the property that for every u , v, c(u) = c(v) = i implies that the distance between u and v is greater than i, is called a k-packing coloring of G. The smallest integer k for which there exists a k-packing coloring of G is called the packing chromatic number of G, and is denoted by χρ(G). Packing chromatic vertex-critical graphs are the graphs G for which χρ(G − x) χρ(G) holds for every vertex x of G. A graph G is called a packing chromatic critical graph if for every proper subgraph H of G, χρ(H) χρ(G). Both of the mentioned variations of critical graphs with respect to the packing chromatic number have already been studied [6, 23]. All packing chromatic (vertex-)critical graphs G with χρ(G) = 3 were characterized, while there were known only partial results for graphs G with χρ(G) = 4. In this paper, we provide characterizations of all packing chromatic vertex-critical graphs G with χρ(G) = 4 and all packing chromatic critical graphs G with χρ(G) = 4.
DOI : 10.2298/FIL2219481F
Classification : 05C15, 05C70, 05C12
Keywords: packing coloring, packing chromatic number, packing chromatic vertex-critical graph, packing chromatic critical graph 
Jasmina Ferme. A characterization of 4-χ ρ -(vertex-)critical graphs. Filomat, Tome 36 (2022) no. 19, p. 6481 . doi: 10.2298/FIL2219481F
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     author = {Jasmina Ferme},
     title = {A characterization of 4-\ensuremath{\chi} \ensuremath{\rho} -(vertex-)critical graphs},
     journal = {Filomat},
     pages = {6481 },
     year = {2022},
     volume = {36},
     number = {19},
     doi = {10.2298/FIL2219481F},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.2298/FIL2219481F/}
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