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In this article, we consider a proper total coloring distinguishes adjacent vertices by sums, if every two adjacent vertices have different total sum of colors of the edges incident to the vertex and the color of the vertex. Pilsniak and Wozniak~\cite{PW} first introduced this coloring and made a conjecture that the minimal number of colors need to have a proper total coloring distinguishes adjacent vertices by sums is less than or equal to the maximum degree plus $3$. We study proper total colorings distinguishing adjacent vertices by sums of some graphs and their products. We prove that these graphs satisfy the conjecture.
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참고문헌 (21)
참고문헌 신청
C. P. de Mello / 2010 / Adjacent-vertex-distinguishing total coloring of indif-ference graphs / Mat. Contemp 39 : 101 ~ 110
C. Thomassen / 2001 / Chromatic graph theory, Challenges for the 21st century / World Sci. Publishing
D. Huang / 2008 / A note on the adjacent vertex distinguishing total chromatic number of graphs / Discrete Math 312 (24) : 3544 ~ 3546
F. Hughes / 1997 / The achromatic number of graphs: a survey and some new results / Bull. Inst. Combin. Appl 19 : 27 ~ 56
G. Dirac / 1952 / Map-colour theorems / Canad. J. Math 4 : 480 ~ 490
C. Thomassen / 2001 / Chromatic graph theory, Challenges for the 21st century / World Sci. Publishing
D. Huang / 2008 / A note on the adjacent vertex distinguishing total chromatic number of graphs / Discrete Math 312 (24) : 3544 ~ 3546
F. Hughes / 1997 / The achromatic number of graphs: a survey and some new results / Bull. Inst. Combin. Appl 19 : 27 ~ 56
G. Dirac / 1952 / Map-colour theorems / Canad. J. Math 4 : 480 ~ 490
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